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June 8, 2009

Algebraic Geometry for Category Theorists

Posted by John Baez

James Dolan and I have spent the last year or so talking about algebraic geometry, trying to learn the basics.

Algebraic geometry should be a lot of fun for category theorists — after all, this is the subject made Grothendieck invent topos theory! But alas, introductions to topos theory don’t seem to explain much about algebraic geometry, and introductions to algebraic geometry don’t seem to fully embrace topos theory. It seems that Grothendieck’s revolution never fully caught on. And that’s sort of sad.

It turns out that algebraic geometry is a lot easier to understand — at least for category theorists — if you grab ahold of the concept of doctrine and take full advantage of it. In algebraic geometry, every space is the moduli space of models of some theory in some doctrine… so the key to understanding algebraic geometry is understanding the doctrines involved.

That last sentence was not supposed to make sense right away! So don’t panic if it didn’t. Now you can see the beginning of a paper where Jim and I will try to explain it. In the reverse of our usual pattern, he’s done all the writing so far:

This page also links to 6 videos of lectures Jim gave on this topic.

With luck, this paper will grow to cover a lot more material. If and when it does, I’ll let you know here at the nnCafé.

Posted at June 8, 2009 12:59 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1987

236 Comments & 1 Trackback

Re: Algebraic Geometry for Category Theorists

I’m very curious about the algebraic geometry, but it seems need a private password…

Posted by: blackball on June 8, 2009 4:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Try it again now — I’ve attempted to fix it. Does it work better now?

Posted by: John Baez on June 8, 2009 4:53 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I’m not sure if it is just my lack of knowledge about media on linux, but I can’t get sound to work on the videos. Can anyone confirm or deny this? The .mov format is not typical for me…

Posted by: Matt on June 8, 2009 4:28 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I believe Windows and Mac users typically use ‘Quicktime’ to view (and hear!) .mov files. That’s what I use, anyway. Alas, I don’t know how this works on UNIX. Can someone else comment?

Posted by: John Baez on June 8, 2009 5:09 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

sudo aptitude install w32codecs
sudo aptitude install ubuntu-restricted-extras

Those will help with Ubuntu, but every flavor of linux has its own syntax. It’s probably easiest to install VLC rather than Quicktime (which may need a divx plugin) on Linux. Also I think Firefox is known to work with VLC. If you still have a problem with this, ask on your Linux flavor forum.

Posted by: Stephen Harris on June 8, 2009 11:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Matt,

I have the same problem. Bummer. I’d really like to see these! I suspect the only fix will be to update to a newer version of linux (I’m still running Fedora 8). If anybody has a better idea, I’d like to know.

Technical stuff:
Ordinarily, I can view Quicktime files just fine, but these ones won’t work for me: apparently, it requires the “amr codec” for interpreting the sound data. This is a non-free codec, but you can download it anyway. Unfortunately, it seems that “ffmpeg” (the package containing the audio/video decoding libraries) isn’t compiled to recognize amr, even if it is on my system.

Posted by: Charles Rezk on June 8, 2009 6:47 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

http://www.penguin.cz/~utx/amr
The problem is proprietary codecs which require a license before you compile them on VLC which is open source.
Because you are academics (not commercial) these will probably be free. I dual boot, and couldn’t play the video on Quicktime Pro on Windows. So I watch the video on VLC
on one smaller window and play the audio on another smaller window, and sync them manually (start video first).
As long as you can play the video on VLC in Linux, then JB and crew should be able to redistribute the audio separately in mp3 format and you could sync them as I did. It seems to me that the app which created the audio portion should also have a ‘save as’ feature for the non-proprietary .mp3 format or there is likely a converter to mp3 from the original audio format. This solution assumes there is a problem in recreating the video download files in another format, redoing all of the released file.

Posted by: Stephen Harris on June 8, 2009 9:13 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I found out the Windows problem with the file was due to Quicktime Pro itself. So substituting Nero Showtime played the file.

Maybe it would help Linux if the original file of the video/audio was saved as “self-contained”. Sort of similar to how when you save a webpage or a latex doc you can choose to include the fonts so that it will display ok even though those particular fonts are not already installed on the target machine.

Posted by: Stephen Harris on June 9, 2009 2:03 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Maybe it would help Linux if the original file of the video/audio was saved as “self-contained”.

i’m not up on the technical issues here but i’ve been trying to get the local technical people to take a look at the problems; i’ll try to see that they get this information.

Posted by: james dolan on June 9, 2009 3:49 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

There are various builds of the free player VLC for *nix platforms, and I’d try those. The audio format that the editing machine has available (and more importantly, works on the streaming server) is AMR, not the more common MP3/MP4. Quicktime plays it fine on various Mac and PC boxes, don’t have access to any other operating system.

Curious, how many are viewing it as a stream (mainstream.ucr.edu/filename.mov) vs downloading it (mediaserve.ucr.edu/filename.mov)?

Posted by: siddiq on June 10, 2009 1:18 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

There are various builds of the free player VLC for *nix platforms, and I’d try those. The audio format that the editing machine has available (and more importantly, works on the streaming server) is AMR, …
————————————–

That has been covered in this thread. There is no precompiled appropriate version of VLC available.

http://www.penguin.cz/~utx/amr

The problem is proprietary codecs which require a license before you compile them on VLC which is open source.
——————————————

You have to recompile VLC for Linux to play these files of Jim Dolan giving lectures. Before that, you have to obtain licenses. VLC on Windows had audio and no video. Only Nero Showtime played the files correctly.

Posted by: Stephen Harris on June 11, 2009 11:23 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Correction, VLC on Windows have no audio
but did have video. Other Windows apps had audio and no Video. Just Nero worked.

Posted by: Stephen Harris on June 11, 2009 11:26 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

A direct link.

Posted by: Jacques Distler on June 8, 2009 4:41 AM | Permalink | PGP Sig | Reply to this

Re: Algebraic Geometry for Category Theorists

Perhaps because my computer has Jim’s password stored on it, your ‘direct link’ takes me to the exact same page as the link I gave. Is yours better for most people?

What I’d really like is a link that could take anyone in the world straight to

http://ncatlab.org/jamesdolan/published/Algebraic+Geometry

I believe this URL does not, in fact, do that job.

Posted by: John Baez on June 8, 2009 4:52 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Both Jacques’s link and

http://ncatlab.org/jamesdolan/published/Algebraic+Geometry

take me to the appropriate pages without any login.

Posted by: Aaron on June 8, 2009 5:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Okay, great — thanks for the speedy response, Aaron! I’ll use this to improve my blog entry.

Posted by: John Baez on June 8, 2009 5:13 AM | Permalink | Reply to this

relation to Lurie/Spivak?

But alas, introductions to topos theory don’t seem to explain much about algebraic geometry, and introductions to algebraic geometry don’t seem to fully embrace topos theory. It seems that Grothendieck’s revolution never fully caught on. And that’s sort of sad.

There is however Jacob Lurie’s structured spaces. I read that as coming pretty close to this dream (or of what I imagine that dream would be).

I’d be interested to learn how you and Jim Dolan think of the approach that you are presenting in relation to that.

In particular, the line objects that I see mentioned on Jim Dolan’s web (under 2. the highbrow story) remind me of the affine line objects that David Spivak in his implementation of Jacob Lurie’s notion of structured space in the smooth context considers in his definition of category of local models (def. 1.1.2, p. 21) refining Jacob Lurie’s notion of geometry (def. 1.2.5, p. 13).

Could it be that what you have in mind with dimensional algebras (I haven’t looked through Jim Dolan’s lecture notes as yet, to be frank) is in turn something like a refinement of David Spivak’s notion of a category of local models?

Posted by: Urs Schreiber on June 8, 2009 12:57 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

james explains that the elements of degree N in a graded commutative algebra correspond to sections of a line bundle, the one pulled back from projective space. This is correct if the graded algebra is generated in degree 1, but more generally, the embedding is in a weighted projective space and the “line bundle” is really an orbibundle, where some fibers are lines and some are [line/cyclic group].

Simplest example: C[x] where x is degree 2. Proj of this is not really a point: it’s a point/Z_2. And the “line bundle” is really line/Z_2. A section of such a thing is really a Z_2-invariant section upstairs. For kth powers of this orbibundle, where k is odd, one is trying to take elements of C that are invariant under negation – so just 0. Whereas for k is even one’s taking elements of C that are invariant under negation^k = identity, so everything. Which of course recovers that C[x] is 0 in odd degree, C in even degree.

Posted by: Allen Knutson on June 8, 2009 1:39 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

james explains that the elements of degree N in a graded commutative algebra correspond to sections of a line bundle, the one pulled back from projective space. This is correct if the graded algebra is generated in degree 1, but more generally, the embedding is in a weighted projective space and the “line bundle” is really an orbibundle, where some fibers are lines and some are [line/cyclic group].

this is already covered in the talks. the concept of “orbibundle” is clumsy, perhaps for several reasons, but mainly because it misses the point that in a model of the dimensional theory over a field, each dimension (aka “line object”) gets realized as a plain old line, not as some sort of mutant quotient space of a line.

roughly it’s like this:

the concept of “bundle over a space x” is (if we temporarily ignore the “cohesive structure” on x and just think of it as a discrete set of points) a special case of the concept of “functor on a category x”; the fiber of the bundle over a point x1 in x is the value of the functor at the object x1 in the discrete category x.

later however people realize that they need the more general case where x, besides being non-discrete as a space (which we’re ignoring here), is also non-discrete as a category; thus x is a topos rather than a mere space. just as before, functors on the category x are still of interest, and those functors still take their values in whatever target category you’re interested in (for example in the category of vector spaces, in the case of “vector bundles”). but if you try to shoe-horn such a functor into the naive “bundle” picture despite the fact that x is now a topos rather than a mere space, then it appears that the “fibers” are now some mutant quotiented-out version of the objects of the target category. so forget the “fibers”; it’s much better to instead think in terms of the values of the functor, which are still just plain ordinary objects of the target category.

(i’m trying (though presumably failing) to out-lawvere lawvere here, in the sense that when lawvere says “space” it’s often somewhat ambiguous as to what category the “space” is supposed to be an object of; here when i say “topos” it’s somewhat ambiguous as to what doctrine the “topos” is supposed to be a theory of.)

Posted by: james dolan on June 8, 2009 7:52 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Is there a definition in simple terms of a Doctrine written anywhere around here? I gather from a previous thread that a doctrine is a monad boosted up one categorical level. If so, then it should be easy to write down the definition in simple 2-categorical terms, without any logic-inspired words like “theory” and “model”. Is it anything subtler than taking the definition of monad and changing all nn-concepts to (n+1)(n+1)-concepts and then writing down some (obvious?) coherence conditions?

Also, there must be a version of Beck’s theorem in this context. Is that easy to state?

Posted by: James on June 8, 2009 2:35 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

TWF200 discusses doctrines.

Posted by: David Corfield on June 8, 2009 2:56 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Why do people say “doctrine” rather than 2-monad? Anyone who hangs out around here should be able to guess what a “2-monad” is even if they’ve never heard of one before, but “doctrine” conveys no intuition (to me) until someone tells you that it means “2-monad.”

Posted by: Mike Shulman on June 8, 2009 4:59 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Mike wrote:

Why do people say “doctrine” rather than 2-monad?

The main reason is outlined in my reply to James below: the word ‘doctrine’ brings its own intuitions, which are valuable in themselves, in part because they haven’t been completely formalized yet. There are things that perhaps should be doctrines that aren’t 2-monads.

The vague idea of a ‘doctrine’ is that it’s a kind of ‘language’ in which you can write theories. Some doctrines are more powerful than others. In a powerful doctrine you can say fancy things; in a weak one you can only say very simple things.

But then something funny happens: if we use category theory to make the concepts of ‘theory’ and ‘doctrine’ precise, we realize that a ‘doctrine’ is like a categorified version of a ‘theory’.

Why? Because more technically, a theory TT is a category equipped with extra stuff! A ‘model’ of this theory is a functor F:TSetF: T \to Set, where FF preserves this extra stuff. The idea is that this functor picks out a set and equips it with extra structure.

So, a theory is a category equipped with extra stuff… but what sort of extra stuff? Well, that’s what your doctrine tells you!

So, we see:

A ‘theory’ is something you use to describe sets equipped with extra structure (or more generally objects in categories with extra structure).

A ‘doctrine’ is something you use to describe categories equipped with extra stuff (or more generally objects in 2-categories equipped with extra stuff).

So a doctrine is like a categorified theory.

Now, I’m sure you know this all very well, Mike — I just felt like explaining it to everyone else. But the point is that there are examples of this pattern that don’t arise from 2-monads — at least, not as easily as you might want. (See below.)

Posted by: John Baez on June 8, 2009 5:24 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Why do people say “doctrine” rather than 2-monad? Anyone who hangs out around here should be able to guess what a “2-monad” is even if they’ve never heard of one before, but “doctrine” conveys no intuition (to me) until someone tells you that it means “2-monad.”

“anyone who hangs out around here” is not a good description of my intended audience; there are fashions of speaking and thinking around here that strike me as unhelpfully complicated.

since you seem to suggest that pre-formal intuition can be useful, i’ll give some of my own pre-formal intuition about doctrines here:

first of all, forget 2-monads; they’re a red herring here.

a doctrine is a 2-category in which it’s appropriate to think of the objects as “theories” and to think of a hom-category hom(x,y) as “the category of interpretations of theory x into theory y”
or as “the category of models of the theory x in the environment y”.

in other words, as james borger put it, the concept of doctrine is all about “logic-inspired words like ‘theory’ and ‘model’”.

the important point is that all you need to proceed with the applications to algebraic geometry is:

doctrine = 2-category

theory of a doctrine = object of a 2-category

model of a theory of a doctrine = morphism out of an object of a 2-category

further discussion of how to sharpen the definition of “doctrine” (or of some substitute word) is of course possible but arguably futile unless preceded by an understanding of the intended examples (the main ones being the doctrine of “dimensional theories”, the doctrine of “multi-dimensional theories”, and the doctrine of “coherently locally ringed toposes”) and more especially of the relevant morphisms between them, reflecting the way in which these doctrines fit into a “hierarchy” or “ladder” of “increasing expressiveness”.

(of the 3 doctrines listed above, the first and third have stable precise definitions; the definition (and name) of the second doctrine is more provisional, but an example of the kind of category that qualifies as a theory of the second doctrine is the category of coherent sheaves over a scheme.)

Posted by: james dolan on June 9, 2009 7:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

there are fashions of speaking and thinking around here that strike me as unhelpfully complicated.

Which ones? Would be nice and helpful to know, and about your ideas on better ways of thinking. We are interested in improving on ourselves by getting rid of bad habits.

Posted by: Urs Schreiber on June 10, 2009 12:23 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

there are fashions of speaking and thinking around here that strike me as unhelpfully complicated.

Which ones? Would be nice and helpful to know, and about your ideas on better ways of thinking. We are interested in improving on ourselves by getting rid of bad habits.

there are too many for it to be possible or useful to list them; i find pretty much everything unhelpfully complicated, and not just here. my comment was very unspecific; my point was just that i don’t subscribe to the general idea that if people have already had to get used to a certain level of complication then that means that there’s no point in striving for a simplification to below that level. my comment was also careless in that reading what mike wrote i don’t see any indication now that he was suggesting otherwise.
Posted by: james dolan on June 10, 2009 1:48 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

James B. wrote:

Is there a definition in simple terms of a Doctrine written anywhere around here?

In one of Jim’s lectures you’ll see a plethora of ways one might try to define the concept of ‘doctrine’. The reason for the plethora is that none seems completely satisfactory as of yet.

Is it anything subtler than taking the definition of monad and changing all nn-concepts to (n+1)(n+1)-concepts and then writing down some (obvious?) coherence conditions?

There is an obvious definition like that, and I bet you can easily guess exactly what it is if I give you some hints.

Lawvere was the first to publish something about doctrines, and for him I believe a doctrine was simply a monad on CatCat, the category of all small categories.

This is good for some purposes, but overly strict for others. So, eventually people tried a subtler concept of ‘doctine’: a pseudomonad on the 2-category of all small categories.

A pseudomonad on a 2-category is the obvious ‘weakened’ version of a monad on a category: the usual associative and left/right unit laws hold only up to 2-isomorphisms — the ‘associator’ and ‘left and right unitors’ — which in turn satisfy coherence laws precisely like those in a monoidal category: the associator satisfies a pentagon identity, and there’s also a law that the unitors must satisfy.

If that sentence didn’t have its desired effect — Oh sure, I know what a pseudomonad is! Duh! It’s obvious! — you can see the definition here:

The definition here is a bit intimidating, perhaps, because it’s been ‘internalized’. Instead of just defining a pseudomonad on a 2-category, Marmolejo defines a pseudomonad in any 3-category. The idea is that a pseudomonad in 3-category 2Cat is the same as a pseudomonad on a 2-category.

And, just to make things even more intimidating, Marmolejo says ‘Gray-category’ where I say ‘3-category’, and ‘Gray’ where I say ‘2Cat’.

But never mind all that! If you just relax and look at the diagrams in his definition, and mentally compare them to the diagrams in the definition for a monad, you’ll see that they are indeed the obvious ‘weakening’.

There’s a very nice theory of pseudomonads. See for example here:

(At least read the abstract!)

However, there are problems with using ‘pseudomonad’ as our notion of ‘doctrine’, since for example there’s no pseudomonad for cartesian closed categories, and one might naively wish to speak of the ‘doctrine of cartesian closed categories’.

Also, there must be a version of Beck’s theorem in this context. Is that easy to state?

I don’t know if such a theorem has been proved here on Earth, though I agree that it must exist in Platonic Heaven.

But I can never even manage to remember the usual version of Beck’s monadicity theorem, so I’m not the one to ask. Maybe Steve Lack will come down and save us.

Posted by: John Baez on June 8, 2009 5:01 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

there’s no pseudomonad for cartesian closed categories

There is; it just lives on the 2-category Cat gCat_g of categories, functors, and natural isomorphisms rather than CatCat.

Also, there must be a version of Beck’s theorem in this context.

One version can be found in

  • Le Creurer, I. J. and Marmolejo, F. and Vitale, E. M. Beck’s theorem for pseudo-monads. JPAA 173 (2002), 293–313.

Another is in the appendix of

  • Hermida, Claudio. Descent on 2-fibrations and strongly 2-regular 2-categories. Appl. Categ. Structures 12 (2004), 427–459.

Note, though, that most definitions of “pseudomonad” require the functor part to be a strict 2-functor. Thus, this is not the “completely weak” version of a 2-monad which would consist of a weak 2-functor on a weak 2-category, etc.

Posted by: Mike Shulman on June 8, 2009 5:26 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Mike wrote:

There is…

You beat me to it, Mike. See below.

Posted by: John Baez on June 8, 2009 5:46 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Thanks for all the pointers and references, everyone.

The paper by Le Creurer, Marmolejo, and Vitale looks to be exactly what I wished for. But you know what they say about wishing for things…

It’s a bit too advanced to help me now. What I think would be really good (for me) is a bunch of examples of 2-monads, where the pseudo-adjunctions are described explicitly. It would also be nice to see how certain free objects can be interpreted as universal operations. (But just in examples! I don’t really care if there’s a 2-version of universal algebra.) For example, at the 1-level, the free G-set on the one element set (i.e., G) can be interpreted as the set of universal unary operations on G-sets.

Are toposes the 2-algebras for a 2-monad on CAT? (TWF 200 seems to suggest so.) If so, can you describe everything explicitly? Are Grothendieck toposes? If not, what goes wrong?

Do abelian categories fit into this picture?

I know all this wasn’t the point of the post, so I wouldn’t object to moving it elsewhere.

Posted by: James on June 9, 2009 2:53 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

One paper you might find helpful is Steve Lack’s 2-categories companion.

Elementary toposes are the algebras for a 2-monad on Cat gCat_g, but, I believe, not one on CatCat. This sort of thing happens whenever you have structure that’s contravariant in one or more of its arguments, like internal-homs and power-objects. Note that the morphisms of algebras for this 2-monad are logical functors, not geometric morphisms.

I believe that Grothendieck toposes are not the algebras for any 2-monad, however, mainly because of size issues.

I don’t recall ever seeing it written down, but I expect that abelian categories are also the algebras for a 2-monad.

Posted by: Mike Shulman on June 9, 2009 6:07 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

By the way, when I say there’s no pseudomonad for CCC’s, I mean there’s no pseudomonad on Cat whose (pseudo)algebras are CCC’s. There is, I believe, a pseudomonad on

Cat g=[categories,functors,naturalisomorphisms]Cat_g = [categories, functors, natural isomorphisms]

whose algebras are CCC’s.

(This has been heavily on my mind lately since I’m working on this topic with Mike Stay.)

So, I’m not saying 2-monads (or better, pseudomonads) are ‘no good’ as a formalization of the intuitive concept of ‘doctrine’. I’m really just trying to say that there’s an intuitive concept there which is a bit tricky to formalize and thus deserves a name different from the various proposed formalizations.

Math is full of words like this: ‘space’, ‘structure’, ‘system’, ‘geometry’, ‘theory’….

So, I don’t say ‘doctrine’ when I know I mean ‘2-monad’ or ‘pseudomonad’. I say it when I want to be a bit vague. And I think James Dolan does that too.

Posted by: John Baez on June 8, 2009 5:44 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

In this discussion and on the nLab page I just created, I’ve been saying “2-monad” in the nLab sense of “2-category,” i.e. (potentially) maximally weak. So “2-monad” in this sense includes “strict 2-monads” (what most people call “2-monads”) and “pseudomonads” of all varieties. Thus, it includes all formalizations of the concept.

What I should have said is that the word “doctrine” conveys no meaning to me prior to hearing an explanation such as your very nice one above. But I think that sort of explanation would make just as much sense if you replaced “doctrine” with “2-monad.” Or, if you want a vaguer word, how about “2-theory?” Just because a 2-dimensional theory is a context in which you can talk about 1-dimensional theories, I don’t see a reason to introduce a new name for it. A 2-category is a context in which you can do 1-category theory, but we still just call it a 2-category.

Also, if we appeal to negative thinking, we see that a 1-theory is a context in which you can talk about 0-dimensional theories. For instance, there is a theory in the context of the 1-theory of groups with two variables xx and yy and one axiom xy=yxx y = y x. A model of this theory in a group GG is two elements of GG which commute; considered as a group itself this theory is the free abelian group on two generators. So I think saying “2-monad” or “2-theory” instead of “doctrine” actually clarifies the idea by pointing out that it isn’t some weird new thing that happens at dimension 2.

Posted by: Mike Shulman on June 8, 2009 6:22 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Mike wrote:

In this discussion and on the nLab page I just created, I’ve been saying “2-monad” in the nLab sense of “2-category,” i.e. (potentially) maximally weak. So “2-monad” in this sense includes “strict 2-monads” (what most people call “2-monads”) and “pseudomonads” of all varieties.

Right… I sort of assumed you were doing that, being a sensible person.

The only reason I called the strict 2-monads ‘2-monads’ and called the weak ones ‘pseudomonads’ is that this is how most of the people who actually work on the subject talk, and I was trying to give a sketchy history of a couple of historically popular attempts to formalize the concept of ‘doctrine’. I’ll quit that now.

Thus, it includes all formalizations of the concept.

It probably doesn’t include the formalization Jim Dolan is looking for, since he knows all this stuff, yet he wrote:

thinking of a doctrine as a 2-monad is definitely on the wrong track here (for reasons that i may get around to explaining).

You can press him for an explanation, if you want.

What I should have said is that the word “doctrine” conveys no meaning to me prior to hearing an explanation such as your very nice one above.

Luckily that phase of your life is over, and henceforth all mathematical terms you encounter will make intuitive sense right away.

But I think that sort of explanation would make just as much sense if you replaced “doctrine” with “2-monad.”

No, because 2-monad means something specific, and right now I’d no more want to restrict the concept of ‘doctrine’ to mean ‘2-monad’ than I’d want to restrict the concept of ‘theory’ to mean ‘monad’.

Or, if you want a vaguer word, how about “2-theory?”

Now you’re talking!

I still have a fondness for ‘doctrine’, and will probably continue to use it out of respect for tradition (which in this case mainly means Lawvere).

But ‘2-theory’ is very nice. It may help people realize that just as a fully general theory lives in a 2-theory, a fully general 2-theory lives in a 3-theory, and so on.

I’ve also got to emphasize: when it comes to this algebraic geometry project, I’m far less interested in the general definition of ‘doctrine’ than in the particular doctrines relevant to algebraic geometry, and how they actually help us think about algebraic geometry!

Sadly, none of the discussion here has yet touched on that issue… or the exciting ideas that Jim explained in what he wrote: the idea that a theory of physics compatible with the tenets of dimensional analysis is actually a presentation of a commutative graded algebra, and that such algebras can be reinterpreted as ‘dimensional theories’, meaning symmetric monoidal categories in which all objects are ‘lines’, and that these in turn give rise to multiprojective varieties (and certain generalizations thereof).

But that’s okay; I’m sure we’ll get to that stuff eventually.

Posted by: John Baez on June 8, 2009 9:45 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

And, just to make things even more intimidating, Marmolejo says ‘Gray-category’ where I say ‘3-category’, and ‘Gray’ where I say ‘2Cat’.

Well, I don’t think Francisco is setting out to intimidate people! :-) He’s just being precise: GrayGray just means 2Cat but with the Gray tensor product as monoidal product (the “right” monoidal product for our purposes here). And a GrayGray-category means a category “enriched in Gray” – a precise way of saying what people in these parts usually mean by a 3-category (as opposed to strict 3-category).

You know this of course, but let’s not give the idea that Marmolejo is being unwarrantedly jargon-y! He’s fine.

Posted by: Todd Trimble on June 8, 2009 7:01 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Of course you’re right. But if I don’t get to joke around, I lose my will to live. And I think most people need all the jokes they can get to survive a close encounter with Gray-categories.



Posted by: John Baez on June 8, 2009 7:09 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Is there a definition in simple terms of a Doctrine written anywhere around here? I gather from a previous thread that a doctrine is a monad boosted up one categorical level.

the closest thing to a formal definition of “doctrine” appropriate for my purposes is at the moment probably:

doctrine = 2-category

theory of a doctrine = object of a 2-category

model of a theory of a doctrine = morphism out of an object of a 2-category

but this is really only a stopgap; i would like a better formal definition but i don’t have one yet. the important point is that the applications of the concept to for example algebraic geometry can and should proceed without having to wait for the perfect definition of “doctrine” to materialize. we have very clear examples of doctrines even if we don’t have a perfect definition of “doctrine” yet.

thinking of a doctrine as a 2-monad is definitely on the wrong track here (for reasons that i may get around to explaining). the idea that “doctrine” should mean (approximately) 2-monad comes from lawvere, but lawvere says that jon beck had an earlier more general definition. lawvere discusses beck’s concept here, for example. as i stated in the lectures, i would like to know more about what beck’s definition actually was, because (whether despite or because of the vagueness of my knowledge of it) it seems closer to what i want than lawvere’s 2-monad definition does.

Posted by: james dolan on June 8, 2009 9:01 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

lawvere discusses beck’s concept here

more specifically, on page 22 at the above link

Posted by: james dolan on June 9, 2009 1:41 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I don’t get much out of Lawvere’s comments at that link, except what John already said. And that still looks to me as though it is well-described by a 2-monad, or at least a finitary 2-monad (aka 2-Lawvere-theory). Or equivalently the category of algebras for such a 2-monad, which of course is equivalent data to the 2-monad if you remember its forgetful functor.

I would like to discuss what you are actually trying to explain, instead of (or in addition to) arguing about terminology, because it sounds really nifty! But unfortunately I don’t have time to watch 6+ hours of video at the moment, and won’t for at least another week.

Posted by: Mike Shulman on June 9, 2009 3:02 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

thinking of a doctrine as a 2-monad is definitely on the wrong track here (for reasons that i may get around to explaining).

i’ll try to get around to it right now:

the problem is that there are important examples of morphisms between doctrines that are not examples of the “natural” (in an informal sense) kind of morphism between 2-monads.

we can see what the natural kind of morphism between 2-monads is by de-categorifying and looking at the natural kind of morphism between monads (that live on a single category s). monads on s are monoids in the monoidal category of endo-functors of s, so arguably the natural morphisms between the monads should be the monoid homomorphisms, and in fact this corresponds to the intuitive concept of “interpretation between algebraic theories” when you think of a monad as an algebraic theory. given such an interpretation f : t1 -> t2 of algebraic theories, we get both a contravariantly induced forgetful functor from the category of t2-algebras to the category of t1-algebras and, if the base category s has appropriate (co-)completeness properties, a left adjoint to the forgetful functor.

but consider the case where s is the category of sets and t1 is the algebraic theory of abelian groups and t2 is the algebraic theory of commutative rings and where we have a “forgetful functor” from the category of t2-algebras to the category of t1-algebras, with a left adjoint going the other way, that _doesn’t_ come from an “interpretation of algebraic theories” in the sense that we’ve been talking about so far. namely, consider the forgetful functor that assigns to a commutative ring its multiplicative group; its left adjoint is the “group ring” functor. thus if you want to count this adjunction as an “interpretation of algebraic theories” then you’ll have to modify your concept of “algebraic theory” and/or your concept of “interpretation of algebraic theories”.

a categorified version of this adjunction is very important in algebraic geometry. a dimensional theory is analogous to an abelian group (because dimensions have inverse dimensions and the tensor product of dimensions is symmetric), and there’s another kind of theory, perhaps we should call it a “multi-dimensional theory” for now, that’s analogous to a commutative ring.

(the category of coherent sheaves over a variety is an example of such a multi-dimensional theory. coherent sheaves are slightly generalized vector bundles, and vector bundles are multi-dimensional objects, in contrast to line bundles which are one-dimensional objects (aka “dimensions”). a category of one-dimensional objects is closed under tensor product and inverse and thus like a categorified abelian group, whereas a category of multi-dimensional objects is closed under tensor product and direct sum and thus like a categorified commutative ring.)

so there’s a forgetful 2-functor from the doctrine of multi-dimensional theories to the doctrine of dimensional theories, assigning to a multi-dimensional theory its subcategory of one-dimensional objects (thus for example assigning to the multi-dimensional theory of coherent sheaves over a variety the dimensional theory of line bundles over the variety). furthermore this forgetful 2-functor has a left adjoint, which is a sort of categorified group ring construction. and this adjunction is (for reasons that hopefully i’ll get around to explaining at some point) of great importance in algebraic geometry. but in analogy to the de-categorified case, we can’t count this adjunction as an “interpretation of doctrines” unless we modify our concept of “doctrine” and/or our concept of “interpretation of doctrines”. that (together with a few similar examples) is my motivation for not being satisfied with defining “doctrine” as “2-monad”.

i should mention that one reason that lawvere’s suggestion to define “doctrine” to mean (approximately) “2-monad” was a somewhat reasonable suggestion is that it does lead to the natural kind of morphism between doctrines being a special kind of adjunction. it’s important (for reasons that i hope to explain) that interpretations of doctrines should be adjunctions, but there are some important adjunctions that don’t qualify as the natural kind of morphism between 2-monads, and that motivates my dissatisfaction with defining “doctrine” to mean “2-monad”.

Posted by: james dolan on June 9, 2009 3:22 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

OK, so the conclusion is that 2-monad (in the maximally weak sense) is not a rough draft of a definition—it is the correct expression of some good platonic concept—but that for your present purposes you need something more general. Is that right?

Posted by: James on June 9, 2009 10:04 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Interesting. I look forward to hearing more, so that I can get some intuition for what a doctrine is (beyond “a 2-category”). Since there is clearly both a 1-version and a 2-version, it’s a bit unfortunate that unadorned “doctrine” seems to be used for the 2-version; it would be nicer if we could call that a 2-doctrine.

This is very vague, but what you’re saying reminds me a bit of toposes. A topos is a category (with extra property-like-structure) and a morphism of toposes is a certain sort of adjunction. A topos can be “presented” in several ways (sheaves on a site, coalgebras for a lex comonad, etc.), but not every morphism between presented toposes comes from a morphism of presentations.

Hmm, let’s take this a bit further. Consider presheaf toposes. Not every geometric morphism between presheaf toposes is induced by a functor, but it is true that every geometric morphism between presheaf toposes is induced by a profunctor. Actually, it should be a profunctor that is lex on one side. Now one might ask the question “what data on a pair of monads induces an adjunction between their categories of algebras”?

I don’t know the answer, but in the one example you mentioned, where T 1T_1 is the monad for abelian groups and T 2T_2 the monad for commutative rings, let T 3T_3 be the monad for abelian monoids; then we do have a span of monad morphisms:

(1)T 1fT 3gT 2.T_1 \overset{f}{\leftarrow} T_3 \overset{g}{\to} T_2.

The induced morphisms of categories of algebras: f f^\star regards a group as a monoid, g g^\star regards a ring as a multiplicative monoid, f !f_! adds inverses to a monoid freely, g !g_! takes the monoid ring, and f f^\star also has a right adjoint f f_\star which takes the group of invertible elements in a monoid. Now the adjunction between T 1T_1-algebras and T 2T_2-algebras you described is the composite adjunction

(2)g !f f g . g_! f^\star \dashv f_\star g^\star.

Just thinking out loud (figuratively speaking–err, writing). Spans are sort of like profunctors….

Posted by: Mike Shulman on June 9, 2009 4:25 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Of course, there is a geometric morphism from the classifying topos of commutative rings to the classifying topos of abelian groups that incarnates the group of units. Presumably the same thing is true for the classifying 2-toposes of the relevant 2-theories. You mentioned classifying toposes at the end of what you wrote; is this what you’re getting at?

Posted by: Mike Shulman on June 9, 2009 5:56 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Of course, there is a geometric morphism from the classifying topos of commutative rings to the classifying topos of abelian groups that incarnates the group of units. Presumably the same thing is true for the classifying 2-toposes of the relevant 2-theories. You mentioned classifying toposes at the end of what you wrote; is this what you’re getting at?

i don’t see what mention of classifying toposes you’re talking about; any mention of toposes by me so far has probably been a “level slip” or so away from what we’re talking about now.

but anyway, you seem to have jumped all the way from suggesting that a doctrine should be a 2-lawvere theory to suggesting that it should be a 2-topos. this is good in some ways; for example, you now seem to be acknowledging that the concept of 2-lawvere theory is too narrow to correspond to the generality of what lawvere described at that link above as “something which is like a theory, except appropriate to be interpreted in the category of categories, rather than, for example, in the category of sets.” and notice how similar your new suggestion is to the suggestion that “doctrine” should mean “2-category”, since if the decategorified case is a reliable guide here then the 3-category of 2-toposes is somewhat similar to the 3-category of 2-categories.

however in jumping from 2-lawvere theory to 2-topos you evidently skipped over the possibility of “2-limits theory” (meaning, roughly, some categorification of the concept of “limits theory”). the point being that, to return to the decategorified case, you don’t have to go all the way to the (geometric) doctrine of toposes to get the concept of “multiplicative group of a commutative ring” to qualify as an interpretation of the theory of abelian groups into the theory of commutative rings; instead you only have to go to the doctrine of (finite) limits theories.

as a matter of fact i think that i’m seeing hints now that for purposes of the algebraic geometry applications that i have in mind, the appropriate concept of “doctrine” may be some sort of categorification of the concept of “limits theory”. part of the intuition behind this is that limits theories are about as far as you can go and still always have a left adjoint to the process of pulling models back along an interpretation of theories. in the categorified case i want to have a left adjoint to the process of pulling 2-models (aka “theories”) back along an interpretation of 2-theories (aka “doctrines”), because my (often unreliable) intuition says that this is a significant aspect of what it means for doctrine d2 to be “richer” or “more expressive” than doctrine d1: that given a theory t of doctrine d1 we obtain by a left adjoint process an “equivalent” theory t’ of the richer doctrine d2, and given a theory e of doctrine d2 we obtain by the corresponding right adjoint process an underlying theory e’ of the poorer doctrine d1, and the adjointness relation “hom(t’,e) = hom(t,e’)” expresses the idea that “what we can express in the poorer doctrine d1 via the theory t we can equivalently express in the richer doctrine d2 via the theory t’”.

(“e” here stands for “environment” which is jargon that i sometimes use for “a theory in co-domain mood, when it wants to be interpreted into instead of out of”.)

another reason (or maybe it’s another way of describing the same reason) for wanting the interpretations of doctrines to correspond to adjunctions between 2-categories of theories is so that sketches (aka “categorified presentations”) of theories can be transported along the left-adjoints (which preserve presentations since these are colimit constructions).

anyway, i still think that there’s only a limited need at this point for ruminating over the abstract concept of “doctrine”, and that ruminating over the relevant specific examples of doctrines is more likely to lead to clarification of the general concept later on.

one thing that puzzles me though is whether the adjointness relation above should be an equivalence of categories or just an equivalence of groupoids. if we (against my general advice) construe the doctrine of dimensional theories as a 2-monad, then i guess that the contravariance of the process of taking the inverse of a dimension pretty much requires that the 2-monad live on a groupoid-enriched category rather than on a category-enriched one. to what extent does this contravariance similarly complicate the task of construing the doctrine of dimensional theories as a “categorified limits theory”? and how does this affect the groupoid-enrichedness vs category-enrichedness of the adjunctions corresponding to interpretations of doctrines? i’m not sure, but perhaps i’m seeing indications that most or all of the “2-categories” relevant to basic algebraic geometry are really just groupoid-enriched categories. i would have to think about this stuff a lot more though to try to decide whether the idle guesses that i’m making now really make any sense.

Posted by: james dolan on June 9, 2009 10:26 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

you now seem to be acknowledging that the concept of 2-lawvere theory is too narrow to correspond to the generality of what lawvere described at that link

I would say rather that 2-monads are just talking about something different. Saying that something is a categorification of a theory, which is basically what Lawvere said, depends of course on what you mean by “theory” in the first place. It seems to me that you’re interested in the “extensional essence” of a (type) theory, as captured by its syntactic category. For a geometric theory, this usually means its classifying topos, and for a finite-product theory, it means its Lawvere theory, and for other things like (finite) limit theories it’s something in between. (I’m sure you know all this, but I’m saying it to get it clear in my head and for anyone else who might be listening.) Adding more structure to the theories (that is, moving to a more expressive 2-theory) allows you more morphisms that you didn’t have before; for instance, abelian groups and commutative rings are both finite-product theories, but (as you said) the morphism you described is not a morphism of finite-product theories, only of finite-limit theories.

So, just as we can talk about the extensional essence of an ordinary theory by using its syntactic category, which is something that lives in the 2-category of models in CatCat of some 2-theory, we now want to talk about the extensional essence of some 2-theory. The 2-categories that you’re saying we can think of as being “doctrines” are then the 2-categories of models in CatCat of some 2-theory (which maybe we could also call the doctrine). Perhaps it is a limit 2-theory, as you suggest. And there are certainly interesting questions involved in making this precise, especially having to do with contravariance. I have some thoughts about how to deal with that, but since this is tangential to the main issue we can stop here for now.

anyway, i still think that there’s only a limited need at this point for ruminating over the abstract concept of “doctrine”, and that ruminating over the relevant specific examples of doctrines is more likely to lead to clarification of the general concept later on.

Unfortunately, I fit John’s definition of a category theorist. In particular, I understand specific examples better after I’ve ruminated over abstract concepts for a while. (-: But I think I have enough to go on now (subject to your next post possibly telling me I’m all wrong, of course). Now if only I had time to watch all your videos.

Posted by: Mike Shulman on June 10, 2009 2:48 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

John’s definition

Sorry, I misspoke: I don’t think that comment was intended as a definition, just an observation.

Posted by: Mike Shulman on June 10, 2009 5:53 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

it does seem a bit ironic to find after this discussion how relatively little disagreement i have with the first paragraph of this that you apparently wrote some time back:

Many structures whose “traditional” definition takes place in Set can be formulated “internally” to any other category (or categorical structure) C with “enough structure.” The structure required on C is often referred to as a doctrine, although it is not necessarily obvious that it will always be a doctrine in the formal sense (that is, a 2-monad).

if i were to revise this nlab entry (which i don’t plan to do because i think that my opinions generally tend to clash too much with other people’s), i might try to make the following points:

1 there are a lot of other categories besides _set_ to which this applies.

2 this informal sense of “doctrine” is the original (category-theoretic) sense, as given by beck. (subject to change if i find information that refutes or modifies this claim.)

3 this original sense of “doctrine” is in some way discernible in lawvere’s discussion of the narrower formal definition that he offered.

4 lawvere has sought to remind people of beck’s prior broader sense of the word.

5 the formal sense of “doctrine” doesn’t seem very useful because “2-monad” seems better for most purposes; in contrast the original informal and still current sense of “doctrine” seems fairly useful.

the most attractive thing for me about the idea of using “2-theory” as a near-synonym for “doctrine” is that it seems even woolier than “doctrine” because of the ungrounded recursion of “an n-theory is a model of an [n+1]-theory”. when i say “doctrine” instead of “2-theory” i’m perhaps trying to encourage myself to at least temporarily stay focused on the microcosm at hand instead of panning outwards towards the surrounding macrocosms. also i don’t strive for extreme consistency in terminology. sometimes i refer to 0-, 1-, and 2-cells in 2-categories as “abstract categories”, “abstract functors”, and “abstract natural transformations”, for example, but i usually don’t call an object of a category an “abstract set” unless i’m trying to make a heavy-handed point.

anyway, i also find in some of your writing places where you say things like “regular, coherent, etc.”. some people got tired of repeating that sort of “et cetera” phrasing and decided to use the informal term “doctrine” instead.

Posted by: james dolan on June 10, 2009 10:53 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

there are a lot of other categories besides set to which this applies.

Do you mean that there are structures whose traditional definition takes place in a category other than SetSet but which can be internalized to any category with sufficient structure? That’s certainly true; I added a comment about it to the “internalization” page.

the formal sense of “doctrine” doesn’t seem very useful because “2-monad” seems better for most purposes; in contrast the original informal and still current sense of “doctrine” seems fairly useful.

Now that I understand something of what the “original informal” sense of “doctrine” is, I agree. I’ve never liked it when people use “doctrine” to mean “2-monad.” I used to just think that meant “doctrine” was a useless word, but now I see that has another different meaning, which makes me believe even more strongly that it should not be used to mean “2-monad.”

I’ve changed the discussion on the nlab pages “internalization” and “2-monad” to accord with this philosophy. There seems to be some weird issue regarding redirection of the page “doctrine,” but once that is worked out we can create the page “doctrine” with a discussion of its informal meaning and various ways that one might try to formalize it.

By the way, the unfortunate fact that by now “doctrine” has been used by lots of people to mean “2-monad” is, I think, now an argument in favor of “2-theory” over “doctrine.” Saying “doctrine” to mean “2-theory” has the potential to create more confusion among people who have encountered “doctrine” as meaning “2-monad” but not in the original sense (like, for instance, me a few days ago).

anyway, i also find in some of your writing places where you say things like “regular, coherent, etc.”. some people got tired of repeating that sort of “et cetera” phrasing and decided to use the informal term “doctrine” instead.

Whenever I write “regular, coherent, etc.” there is a very specific finite list of words that I have in mind (usually, something like “lex, regular, coherent, first-order/Heyting, disjunctive/extensive, geometric”). But it seems that “doctrine” is supposed to be much more widely applicable. Dimensional theories, for instance, are evidently supposed to be a doctrine, but when I write “regular, coherent, etc,” dimensional theories are not one of the things I have in mind.

Posted by: Mike Shulman on June 11, 2009 1:52 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Do you mean that there are structures whose traditional definition takes place in a category other than Set but which can be internalized to any category with sufficient structure?

yes; to give a relevant example, the definition of “line equipped with 3 linear functionals” takes place originally in the category of lines over a field but can be internalized to be interpretable in any dimensional category.

Posted by: james dolan on June 11, 2009 12:23 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

By the way, the unfortunate fact that by now “doctrine” has been used by lots of people to mean “2-monad” is, I think, now an argument in favor of “2-theory” over “doctrine.” Saying “doctrine” to mean “2-theory” has the potential to create more confusion among people who have encountered “doctrine” as meaning “2-monad” but not in the original sense (like, for instance, me a few days ago).

i’m not into encyclopedization or terminology standardization; i’m going to keep using “doctrine” for now (in part because “2-theory” sounds like terminology that i’d rather save for some more systematic usage after the situation gets clarified, or else for some even less systematic usage).

Posted by: james dolan on June 11, 2009 12:42 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I started creating an nlab page for doctrine, based on what I have just learned from this discussion. It is clearly very incomplete.

Posted by: Mike Shulman on June 11, 2009 3:49 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

any mention of toposes by me so far has probably been a “level slip” or so away from what we’re talking about now.

i should mention toposes a lot more, though, because of the significant role that they play in the approach to algebraic geometry that i’m trying to develop. i’ll try to explain something about this now.

consider some nice projective algebraic variety, say for example the projective plane. the projective plane is the space of 1-dimensional subobjects of the vector space k^3, where k is the base field that we’re working over. if we recall the official category-theoretic definition of “subobject of x” as an isomorphism class of monomorphisms into x, then we see directly that the projective plane is thus “the moduli space of 1-dimensional vector spaces equipped with a linear embedding into k^3”.

thus we’re implicitly talking about a “theory” here, namely “the theory of a line object equipped with a linear embedding into 1+1+1” (where “1” denotes the trivial line object and “+” denotes direct sum). but to make this precise we must specify a particular doctrine and check that its syntax is expressive enough to actually express this theory.

so for example can we express this theory in the doctrine of dimensional theories? no, because the syntax of dimensional theories lacks any means of saying that a morphism should be an embedding.

we can come close, however: we can take “the theory of a line object equipped with a linear map to 1+1+1” (or more precisely, “the theory of a line object equipped with linear maps x,y,z to 1”).

then by the theorem that says that a dimensional theory is equivalent to a kind of graded commutative ring, we obtain from this theory a graded commutative ring known as “the homogeneous coordinate ring of the projective plane”.

also, however, if we take any doctrine that’s “more expressive” than the doctrine of dimensional theories, then we can automatically promote our dimensional theory to a theory of the more expressive doctrine. an example of such a more expressive doctrine is the doctrine of “coherently locally ringed toposes”. a theory of this doctrine is an “extension” of the coherent geometric theory of local rings. a “local ring” is something that’s a lot like a field, enough so that you can talk about the moral equivalent of vector spaces over it, and this is what makes this doctrine more expressive than the doctrine of dimensional theories- anything that you can say in a dimensional theory, you can paraphrase in an extension of the coherent geometric theory of local rings. but in fact the new doctrine is _strictly_ more expressive than the previous one, because for example we can easily express in the new doctrine the idea that a linear map is monic. thus in this new doctrine we can actually fully express the theory of “a line object equipped with a linear embedding into 1+1+1”. the resulting coherently locally ringed topos is called “the big zariski topos of the projective plane”.

there’s another doctrine, intermediate between the doctrine of dimensional theories and the doctrine of coherently locally ringed toposes, in which the theory of “a line object equipped with a linear embedding into 1+1+1” can also be fully expressed. the theory as expressed in this intermediate doctrine is sometimes called “the category of coherent sheaves over the projective plane”.

i’ll have to try to flesh this story out later (and try to correct some of the mistakes that i’ve probably made). part of the moral is supposed to be that for a variety (or scheme or …) x, things like the “category of coherent sheaves over x” or the “big zariski topos of x” can be described in a more conceptual way than they often are.

Posted by: james dolan on June 10, 2009 7:34 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Why do we still get the (or “a”) right answer in the case of dimensional algebras, even though we can’t express the fact that the morphism should be an embedding? Does this have to do with the “irrelevant ideal” in a homogeneous coordinate ring?

Posted by: Mike Shulman on June 11, 2009 3:16 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Why do we still get the (or “a”) right answer in the case of dimensional algebras, even though we can’t express the fact that the morphism should be an embedding? Does this have to do with the “irrelevant ideal” in a homogeneous coordinate ring?

yes, exactly. the irrelevance of the irrelevant ideal is in fact highly questionable. in this example it corresponds to the essentially unique model where the map is not an embedding. this degenerate model can be thought of as the fixed point of the “rescaling group” or “renormalization group” given as the “algebraic fourier dual” (sorry, the proper name for that has slipped my mind if i ever knew it) of the grading group. being fixed under the renormalization group, this model has lots of automorphisms, so if you include this point then the moduli space is really a “moduli stack” (meaning that the fiber of the locally ringed topos over a particular local ring is non-localic). presumably this scared algebraic geometers of an earlier era enough for them to declare the ideal “irrelevant”, whereas the fixed point of a renormalization group is in fact highly relevant for some purposes.

when you get to the doctrine where the category of coherent sheaves live, the axiom that says that the map should be an embedding can be expressed by “modding out by a serre ideal subcategory” (which is sort of the analog of “applying a grothendieck topology” in the next doctrine up the ladder, where the big zariski topos lives).

my personal take on it is that the presence of the degenerate models at the level of the dimensional theories is more of a feature than a bug; after all fixed points of renormalization groups are of significant interest in dimensional analysis. i could imagine someone trying to re-design the doctrine so as to somehow eliminate the degenerate models in a less ad hoc way than by just declaring certain models “irrelevant”, but i never saw any way to accomplish that myself. the degenerate models can optionally be removed by adding the embedding axiom to the theory in the higher doctrines but leaving them in is sometimes more fun.

Posted by: james dolan on June 11, 2009 5:39 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

my personal take on it is that the presence of the degenerate models at the level of the dimensional theories is more of a feature than a bug; after all fixed points of renormalization groups are of significant interest in dimensional analysis.

of course “fixed points of renormalization groups” is an overblown phrase in this context, but i think that it’s suggestive in an interesting way.

Posted by: james dolan on June 11, 2009 5:48 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

being fixed under the renormalization group, this model has lots of automorphisms, so if you include this point then the moduli space is really a “moduli stack” (meaning that the fiber of the locally ringed topos over a particular local ring is non-localic).

roughly speaking, the local ring is being equipped with “stuff” rather than mere “structure”.

Posted by: james dolan on June 11, 2009 6:50 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

You just used a whole lot of words that I hope to understand one day, but I’m happy that the answer is “yes.”

Posted by: Mike Shulman on June 11, 2009 3:34 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

You just used a whole lot of words that I hope to understand one day, but I’m happy that the answer is “yes.”

if there’s a particular word (or even a particular bunch of words) there that seems opaque to someone, then i can try to provide some understanding of it if you let me know which one. i did try to explain more about some of the words over here.

Posted by: james dolan on June 11, 2009 10:59 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I don’t understand what is meant by “renormalization group.” I get that in this case it just means “rescaling,” but why do you call it “renormalization?”

Posted by: Mike Shulman on June 12, 2009 6:01 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I don’t understand what is meant by “renormalization group.” I get that in this case it just means “rescaling,” but why do you call it “renormalization?”

it’s a joke, but the kind that could actually turn serious at some point. as far as i know “rescaling” and “renormalization” are pretty much synonyms, and riffing on them is a way for me to remind myself to look for connections between the renormalization groups that are showing up here and the renormalization groups that play a significant role in mathematical physics.

Posted by: james dolan on June 12, 2009 6:46 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

as far as i know “rescaling” and “renormalization” are pretty much synonyms,

Renormalization denotes the behaviour of certain systems under a rescaling. The rescaling is a simple linear transformation, the renromalization-reaction of the system to that is in general complicated.

More precisely, there are families of quantum field theories on Minkowski space parameterized by a list of numbers called coupling constants. It so happens that one of these sets of parameters corresponds to another set of these parameters if at the same time an overall scale factor of the Minkowski metric is applied.

Conversely. starting with one set of parameters and then changing the scale parameter produces a 1-dimensional curve in the space of coupling constants of the QFT. This is called “the running of the coupling constants” and the flow it induces on all of coupling constant space is called the “renormalization group flow”.

This is in general given by a non- trivial equation.

For instance certain families of 2-dimensional theories have a continuum of coupling constants and the renormalization flow of them is rich enough to exhibit the phenomenon that its critical points alone are in bijection to solutions of Einstein’s equations of gravity in a higher dimensional space.

So I’d say that “renormalization” is something that involves “reparameterization” but is a considerably richer phenomenon: it is the reaction to rescaling of a system.

Posted by: Urs Schreiber on June 13, 2009 10:06 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

My understanding had always been that we call it the irrelevant ideal because WE CHOOSE to consider the corresponding subfunctor as irrelevant. In your example of the projective plane, your theory has a number of subfunctors: the zero subfunctor (assigns the empty set to every ring), the maximal ideal (the set of maps from a line object to 1+1+1 which are identically zero), the square of the maximal ideal and so forth.

If we decide that < x,y,z > is irrelevant, then we are deciding that the images of these subfunctors are isomorphic to the zero object. I suspect that the 2-category way of saying this is that we have adjoined a 2-isomorphism between these functors and the zero functor. This is actually one of my favorite examples of how adjoining isomorphisms can change a category: the category of (finitely generated) graded k[x,y,z] modules is slightly different from the category of (coherent) sheaves on P^2.

Of course, you may not WANT to make those subfunctors are irrelevant. I have often had a problem get much easier by working in the category of graded modules. But I think the terminology is very good.

Posted by: David Speyer on June 15, 2009 2:08 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

My understanding had always been that we call it the irrelevant ideal because WE CHOOSE to consider the corresponding subfunctor as irrelevant. In your example of the projective plane, your theory has a number of subfunctors: the zero subfunctor (assigns the empty set to every ring), the maximal ideal (the set of maps from a line object to 1+1+1 which are identically zero), the square of the maximal ideal and so forth.

If we decide that is irrelevant, then we are deciding that the images of these subfunctors are isomorphic to the zero object. I suspect that the 2-category way of saying this is that we have adjoined a 2-isomorphism between these functors and the zero functor. This is actually one of my favorite examples of how adjoining isomorphisms can change a category: the category of (finitely generated) graded k[x,y,z] modules is slightly different from the category of (coherent) sheaves on P^2.

Of course, you may not WANT to make those subfunctors are irrelevant. I have often had a problem get much easier by working in the category of graded modules. But I think the terminology is very good.

the category of fg graded k[x,y,z]-modules also deserves to be called the category of coherent sheaves on something, namely on “the moduli stack for a line l possessing points x,y,z”. the projective plane is, in contrast, “the moduli stack for a line l generated by points x,y,z”. the slight difference is that in the latter case, the map “x+y+z” from the coherent sheaf 1+1+1 to the coherent sheaf l has had its cokernel demoted to zero, or equivalently the inclusion of its image promoted to an isomorphism.

different models or different ideals are irrelevant for different purposes. the meaning of the term “the irrelevant ideal” has become calcified from focusing on one possible purpose over others.

part of my viewpoint is that the doctrine of dimensional theories is worth studying in itself, not just as an auxiliary to the doctrine where categories of coherent sheaves live. in the doctrine of dimensional theories you _don’t_ get to choose to make some models irrelevant by demoting some cokernel to zero, because the syntax of the doctrine simply lacks that expressiveness; the process of taking cokernels generally takes you out of the world of line objects and into the world of coherent sheaves.

another part of my viewpoint is that you get very beneficial conceptual simplifications from thinking directly in terms of theories instead of indirectly in terms of their moduli stacks.

Posted by: james dolan on June 16, 2009 10:31 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

the category of fg graded k[x,y,z]-modules also deserves to be called the category of coherent sheaves on something, namely on “the moduli stack for a line l possessing points x,y,z”. the projective plane is, in contrast, “the moduli stack for a line l generated by points x,y,z”. the slight difference is that in the latter case, the map “x+y+z” from the coherent sheaf 1+1+1 to the coherent sheaf l has had its cokernel demoted to zero, or equivalently the inclusion of its image promoted to an isomorphism.

“x+y+z” seems like bad notation for the map of coherent sheaves that i was trying to describe above; better might be something like “(x,y,z)” or “(a,b,c)|->ax+by+cz”.

Posted by: james dolan on June 20, 2009 11:37 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

responding to david speyer’s comments, i wrote:

the category of fg graded k[x,y,z]-modules also deserves to be called the category of coherent sheaves on something, namely on “the moduli stack for a line l possessing points x,y,z”.

i have a question about this stack: is it a deligne-mumford stack, or just an artin stack?

i’m in the process of learning about “deligne-mumford stacks” and “artin stacks” at the moment, as discussed for example in the wikipedia article on algebraic stacks and instead of trying to decipher the fine print of a formal definition, it seems like it would be easier for me to grasp what these kinds of stacks are by asking a few questions about examples like this one.

it suggests at the link above that an algebraic stack qualifies as an artin stack if the “stabilizer groups” are algebraic groups. since the stack that i’m asking about is the orbit stack of a representation of the algebraic group gl(1) (namely the direct sum of 3 copies of the tautological representation), it seems pretty much of a shoo-in to qualify as an artin stack. (gl(1) acts as the re-scaling group on linear 3-space with the origin as unique fixed point, so the only non-trivial stabilizer subgroup is gl(1) itself.)

it also suggests there that an algebraic stack qualifies as a deligne-mumford stack if the stabilizer groups are “finite-dimensional”. gl(1) sure strikes me as being “finite-dimensional”, so i’ll guess that this stack qualifies as a deligne-mumford stack too.

except that there seem to be some other hints in what they say there that this stack is _not_ a deligne-mumford stack, and that “finite-dimensional” in this context really means something more like “finite”.

for example the original application to moduli stacks of algebraic curves doesn’t especially require non-finite stabilizer subgroups. also the projection from linear 3-space to the orbit stack doesn’t seem very “etale”, since it has 1-dimensional fibers. but maybe there’s some other map from a scheme to this stack that’s etale (as required by the definition given at the link for “deligne-mumford stack”)?

so is the stack in question a deligne-mumford stack, or not?

Posted by: james dolan on July 7, 2009 1:01 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

James,
The stack in question (the quotient of affine 3-space by the multiplicative group) is certainly Artin (being a quotient of a variety by an algebraic group action), but not Deligne-Mumford. DM stacks always have stabilizer groups which finite (i.e., the ring of functions on them is finite dimensional over the ground ring): as you point out, DM stacks are quotients of varieties by etale groupoids, which can only have finite stabilizers.

One of the nicest important examples of an Artin stack is even simpler, namely the quotient of just the affine line by the multiplicative group. It has the nice feature that coherent sheaves on it are the same as filtered vector spaces. Simpson has used this picture to define what it means to have a filtration on all kinds of things, like on schemes or stacks or categories or higher categories or… - it just means that you have a scheme/category/etc. over this quotient stack, or equivalently you have one over the affine line, equivariantly for the action of the multiplicative group. This is how one defines eg the Hodge filtration on nonabelian cohomology.

Posted by: David Ben-Zvi on July 7, 2009 4:07 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

thanks david for your answers here.

DM stacks always have stabilizer groups which finite (i.e., the ring of functions on them is finite dimensional over the ground ring)

i think that the reason that i was so unwilling to believe that “finite-dimensional” is in reference to the ring of functions on the group here is that i’m more used to thinking about algebraic groups in general (which are not well-characterized by their rings of functions) than about affine algebraic groups, but perhaps part of the point is that affine algebraic groups are adequate for the purpose here because being deligne-mumford is in some sense an essentially “local” phenomenon. well, maybe that’s not quite the right way to put it; i’ll have to think some more about why affine algebraic groups seem to be adequate here.

One of the nicest important examples of an Artin stack is even simpler, namely the quotient of just the affine line by the multiplicative group. It has the nice feature that coherent sheaves on it are the same as filtered vector spaces.

thanks especially for this information, since it _doesn’t_ match what my calculations seem to have been telling me so far. i’ll have to think about it some more to try to figure out whether i’m just having some more terminological confusion here or whether instead i have something more seriously screwed up.
Posted by: james dolan on July 7, 2009 5:19 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I wouldn’t say that having affine stabilizer groups was a priori part of the discussion, rather it’s the finiteness - but the algebro-geometric way to define finiteness is via finite dimensionality of rings of functions..

I should point out also that all the Artin stacks that I’ve ever seen arising naturally have the property of having affine diagonal - this means essentially (and certainly implies) that all the stabilizer groups of points are affine algebraic groups (technically, we’re asking for the diagonal map from X to X x X to be an affine morphism). In other words, it’s rare to find a moduli problem where the automorphisms of an object are an abelian variety (unless say you’re studying moduli of genus one curves and unwilling to or unable to fix additional data), and other algebraic groups tend to be affine, so this is a very reasonable condition. It has lots of pleasant consequences, in particular guaranteeing that the theory of coherent sheaves on your stack is reasonable. For example, a paper of Lurie’s shows that such stacks are canonically determined by their categories of quasicoherent sheaves as symmetric monoidal categories - i.e., they are “affine in a categorified sense”.

Posted by: David Ben-Zvi on July 7, 2009 5:44 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I wouldn’t say that having affine stabilizer groups was a priori part of the discussion, rather it’s the finiteness - but the algebro-geometric way to define finiteness is via finite dimensionality of rings of functions..

sorry, i have a pretty stupid question here: the ring of functions on an abelian variety is finite-dimensional (for the “pathological” reason that the variety is non-affine, meaning that the ring doesn’t adequately represent it), so are you saying that an abelian variety actually qualifies as “finite” in the sense that we’re talking about? that would strike me as peculiar offhand.

I should point out also that all the Artin stacks that I’ve ever seen arising naturally have the property of having affine diagonal - this means essentially (and certainly implies) that all the stabilizer groups of points are affine algebraic groups (technically, we’re asking for the diagonal map from X to X x X to be an affine morphism). In other words, it’s rare to find a moduli problem where the automorphisms of an object are an abelian variety (unless say you’re studying moduli of genus one curves and unwilling to or unable to fix additional data),

as a matter of fact i’m occasionally interested in this sort of moduli stack. but to continue my stupid question from above: if abelian varieties qualified as “finite”, then the moduli stack of unmarked genus one curves would meet the criterion of having “finite” stabilizer groups, which is the rule-of-thumb criterion that the wikipedia article gave for qualifying as a deligne-mumford stack. but it’s not deligne-mumford, is it?

the moduli stack of unmarked genus zero curves still wouldn’t be deligne-mumford anyway, though.

and other algebraic groups tend to be affine, so this is a very reasonable condition. It has lots of pleasant consequences, in particular guaranteeing that the theory of coherent sheaves on your stack is reasonable. For example, a paper of Lurie’s shows that such stacks are canonically determined by their categories of quasicoherent sheaves as symmetric monoidal categories - i.e., they are “affine in a categorified sense”.

i may try to say/ask more about this later; there’s a lot to say/ask about it.

Posted by: james dolan on July 8, 2009 8:08 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

The definition of “finite” presupposes affine - finite schemes over a ring are specs of “finite dimensional” algebras over the base (technically, algebras which are finitely generated as modules over the base). So abelian varieties are not finite. In particular DM stacks have affine (in fact etale) diagonal.. The moduli of (unmarked) smooth projective curves is thus DM unless we’re in genus zero (where it still has affine diagonal) or one (where it doesn’t), but always Artin.

Posted by: David Ben-Zvi on July 8, 2009 3:08 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

One of the nicest important examples of an Artin stack is even simpler, namely the quotient of just the affine line by the multiplicative group. It has the nice feature that coherent sheaves on it are the same as filtered vector spaces.

thanks especially for this information, since it _doesn’t_ match what my calculations seem to have been telling me so far. i’ll have to think about it some more to try to figure out whether i’m just having some more terminological confusion here or whether instead i have something more seriously screwed up.

let me try to describe my terminological confusion here. for the orbit stack of the tautological representation of gl(1), the thing that i would call the category of quasicoherent sheaves over it is the category of graded modules of the graded algebra of one-variable polynomials (with its usual z-grading).

i don’t know the precise details of any standard definitions of “orbit stack” and “quasicoherent sheaf”; instead i’ve just been using my own definitions. if it turns out that there’s some serious conflict between my definitions and standard definitions then i might have to change mine.

but perhaps there isn’t any serious conflict here. the category of graded modules in this example is essentially just the representation category of the quiver “…->.->.->.->…”. the (z-)filtered vector spaces (according to the usual definition of “filtered” that i’m familiar with) can be thought of as the projective such representations. so perhaps that’s what you’re saying?

Posted by: james dolan on July 8, 2009 11:35 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I was being sloppy - as you point out one should consider vector bundles on the line - or (equivalently in this case) ask for your coherent sheaves to be flat over 0 - in other words we have to avoid torsion sheaves at 0 (since we can stick in any graded vector space as a GL1 equivariant sheaf on the point). In any case, the Rees construction takes a filtered module and makes of it a graded module over polynomials in one variable, i.e. a sheaf on the stack in question, and we can reverse the process given the desired flatness at 0.

Posted by: David Ben-Zvi on July 8, 2009 3:14 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

hmm, and i guess that the process of pulling a vector bundle back along the map between orbit stacks induced by the zero intertwiner from the trivial representation of gl(1) to the tautological representation is essentially the process of taking the associated graded vector space of a filtered vector space.

Posted by: james dolan on July 9, 2009 2:28 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

which suggests some ideas about the role of the “associated graded vector space” construction in “deformation theory”, though not clearly enough for me to say anything very interesting about it yet.

Posted by: james dolan on July 9, 2009 3:33 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

perhaps i should at least say that there’s a better answer to mike shulman’s question lurking here than i managed to give over here. that is, there are relationships (which i wish i understood a lot better) between renormalization group ideas in physics and deformation theory which help to explain why it might be interesting to think of the group gl(1) as it occurs in this discussion as a “renormalization group”.

Posted by: james dolan on July 9, 2009 5:58 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

for anyone trying to follow along here, the definition of “vector bundle over a stack s” in this context is evidently something like “projective object in the category of quasicoherent sheaves over s”, which makes sense in keeping with the philosophy of the serre-swan theorem.

i hope to eventually say more here about what quasicoherent and/or coherent sheaves are and the conceptual role that they play, though.

Posted by: james dolan on July 10, 2009 9:50 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

The Serre-Swan theorem, characterizing vector bundles as projective modules over functions, only applies in affine situations such as affine schemes or C C^\infty-manifolds (which are cohomologically affine, thanks to partitions of unity). For example the trivial vector bundle is not projective whenever you have sheaf cohomology, e.g. on projective space, and categories of sheaves don’t tend to have many projectives. Vector bundles are locally projective, or locally free, however.

To define a vector bundle or other kinds of sheaves on a stack or scheme, one can use the fact that sheaves are defined by their local properties, and schemes and stacks are (by definition) locally affine schemes (i.e. locally given by a ring). Thus we define a quasicoherent sheaf as one that’s locally (on affines SpecR\operatorname{Spec} R) just a module over the corresponding ring RR, a coherent sheaf as one that’s locally a finitely presented RR-module, and a vector bundle as a sheaf that’s locally a projective module (or, even more locally, a free module).

Posted by: David Ben-Zvi on July 13, 2009 6:12 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

The Serre-Swan theorem, characterizing vector bundles as projective modules over functions, only applies in affine situations such as affine schemes or C ∞-manifolds (which are cohomologically affine, thanks to partitions of unity). For example the trivial vector bundle is not projective whenever you have sheaf cohomology, e.g. on projective space, and categories of sheaves don’t tend to have many projectives. Vector bundles are locally projective, or locally free, however.

the serre-swan theorem does seem to hold though for those artin stacks arising as moduli stacks of dimensional theories; roughly, the orbit stacks of torus actions on affine schemes. the category of quasicoherent sheaves over such a stack is just a category of graded modules, so there’s no shortage of projective objects in it; in particular the trivial line bundle is just a free module.

for example consider the orbit stack of the scaling action of gl(1) on a vector space v. the quasicoherent sheaves over the orbit stack are the graded modules of the z-graded algebra of polynomial functions on v, and the trivial line bundle is the free module on one generator in grade zero; so the trivial bundle is projective and there’s no sheaf cohomology.

but now consider the projective space p(v) as a substack of this orbit stack. the whole orbit stack is the moduli stack for the dimensional theory of “a line equipped with a linear map from v”; the substack is carved out by the extra axiom that the image of the map should generate the line. expressed as a “covering condition”, this axiom says that a certain manifestly non-surjective homomorphism of graded modules is nevertheless epi in the category of quasicoherent sheaves over the substack. that seems like a direct indication of the sheaf cohomology of p(v): an epi that’s non-surjective on global sections, thus witnessing the non-projectiveness of the trivial line bundle.

according to my calculation, this is saying that there’s a bundle of affine vector spaces over p(v) where the fiber over a 1d subspace l of v is the space of retractions from v onto l, which is a torsor of the vector space hom(v/l,l), but which has no section and is thus globally non-trivial, thus giving a non-trivial element of the first sheaf cohomology group of the vector bundle “l |-> hom(v/l,l)”.

i’m guessing that, in some vague sense that i’m not prepared to defend at the moment, this element should “generate” the entire coherent cohomology of p(v).

this suggests that the omission of “the irrelevant point” from the moduli stack of a dimensional theory is double-edged. on the one hand it turns (for example) the nice, simple, and important example of an artin stack that david mentioned into boring old projective 0-space, but on the other hand it makes non-trivial coherent cohomology possible.

To define a vector bundle or other kinds of sheaves on a stack or scheme, one can use the fact that sheaves are defined by their local properties, and schemes and stacks are (by definition) locally affine schemes (i.e. locally given by a ring). Thus we define a quasicoherent sheaf as one that’s locally (on affines SpecR) just a module over the corresponding ring R, a coherent sheaf as one that’s locally a finitely presented R-module, and a vector bundle as a sheaf that’s locally a projective module (or, even more locally, a free module).

there’s also a different philosophy according to which it’s more useful to take the category of coherent (or quasicoherent) sheaves as primary and to define the corresponding stack in terms of it; i hope to say more about this philosophy later.

Posted by: james dolan on July 16, 2009 11:11 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i want to try to describe a bit more clearly here the contrast between the whole orbit stack of the scaling action of gl(1) on a finite-dimensional vector space v, which is a non-trivial artin stack with trivial coherent cohomology, and the substack p(v) = the projective space of v, which is a mere scheme but with non-trivial coherent cohomology.

i said above that the whole orbit stack is the moduli stack of the dimensional theory of “a line l equipped with a linear map from v”, but that was a slip; instead it should have been a linear map _to_ v (or equivalently, from v* to l*). then a model of the more restrictive (and not merely dimensional) theory satisfies the extra axiom that the map to v is an embedding.

in this context being an embedding is equivalent to having a retraction, so the points in the substack p(v) are precisely the points in the whole orbit stack for which the retractions of l->v form a torsor of a vector space. (a torsor must be non-empty.)

this shows more clearly how we get non-trivial coherent cohomology over the substack p(v) but not over the whole orbit stack, as the torsor that gives a non-trivial cohomology element exists only over the substack.

Posted by: james dolan on July 17, 2009 7:59 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

sorry, i think that i found mistakes in the calculations that i used in the above two posts. i may try to straighten the situation out later.

Posted by: james dolan on July 18, 2009 1:18 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i’ll have to try to flesh this story out later (and try to correct some of the mistakes that i’ve probably made). part of the moral is supposed to be that for a variety (or scheme or …) x, things like the “category of coherent sheaves over x” or the “big zariski topos of x” can be described in a more conceptual way than they often are.

i want to say some more here about the concept of “big zariski topos” and how topos theory fits into algebraic geometry, indeed “fleshing out” what i said about it above. i hope that people will correct me when i make mistakes, and i’d be interested to hear alternative points of view.

as with much of the exposition here, this is primarily directed towards people who would like to use their understanding of category theory to better understand algebraic geometry; in particular it helps to have some understanding of various doctrines (for example the doctrine of finite products theories, or of finite limits theories, or of coherent geometric theories, and so forth), though not necessarily of a formal concept of “doctrine”.

(coherent geometric theories are also known as “coherent toposes”. understanding how to view toposes as theories requires a fair amount of background, such as discussed in section d “toposes as theories” of johnstone’s “sketches of an elephant, a topos theory compendium”.)

when i say that i’m going to try to explain “how topos theory fits into algebraic geometry” i mean at a very foundational level, rather than at the level of specific applications such as using etale toposes to prove the weil conjectures or using nisnevich toposes to study so-called “motives”. i’d very much like to understand such applications but i don’t understand them well enough yet to say much about them, except to say that there’s a conceptual unity between the different kinds of toposes used in algebraic geometry (zariski, etale, nisnevich, and so forth, including “big” and “small” versions of all of these), so that even though i’m going to focus on the case of the big zariski toposes, almost all of what i say here is relevant to the other kinds of toposes as well.

the main thing that i want to try to explain here is the process that associates to a dimensional theory t a topos z(t), called “the big zariski topos of t”.

there’s a standard account of the big zariski topos of a scheme that i find misleading, against which i want to contrast what i find to be a more conceptual explanation. the standard account goes something like this: first define the big zariski topos of an affine scheme as the classifying topos for the theory of local k-algebras, where k is the corresponding commutative ring; then apply the usual philosophy of extending from the case of affine schemes to the case of general schemes by “glueing together” the affine pieces in an appropriate way.

in a later post i hope to discuss more fully the pros and cons of this usual reductionist philosophy of studying general schemes by dissecting them into affine pieces, the cons arguably including what was discussed here: that as the “dimensionless” special case of dimensional theories, affine schemes are atypically boring theories; whereas a typical theory is built up in layers of stuff, structure, and properties, affine theories lack the “stuff” layer (the dimensions), and are thus more like propositional theories than like predicate theories.

however, my main complaint against the standard account of the big zariski topos is of a different nature: that it’s presented as a theory of _the wrong doctrine_. the wrong doctrine is the doctrine of coherent toposes, aka “coherent geometric theories”; the right doctrine is the doctrine of coherently locally ringed toposes, aka “coherent extensions of the geometric theory of local rings”.

a coherent extension of the geometric theory of local rings is like an individualized document written on a standard pre-printed form; the standard pre-printed part is the theory of local rings. if you make the mistake of focusing on the standard pre-printed part instead of on the individualized part, then all of the z(t)’s sound pretty similar; regardless of the choice of the dimensional theory t they all just sound like minor variations on the theory of local rings.

instead you should mostly ignore the standard pre-printed part of z(t) and focus on the individualized part. i’m not sure exactly what the appropriate software engineering metaphor would be these days, but it’s very relevant. the idea is that the standard pre-printed part of z(t) for a dimensional theory t is just there to implement the concept of “dimension”, and moreover implement it in only a contingent way- contingent upon the specification of a specific local ring for the dimensions to be invertible modules of. then the individualized part of z(t) just re-states the original dimensional theory t, invoking the contingently implemented concept of dimension as the need arises.

thus the big zariski theories z(t) are as varied and as different from each other as are the dimensional theories t that they come from; in fact, they _are_ the dimensional theories that they come from, just translated into a richer doctrine. thus for example if t is the dimensional theory of some kind of lie algebra (as discussed here) then z(t) is the theory of that same kind of lie algebra, but over a local ring r, thus giving a coherent extension of the geometric theory of a local ring r.

similarly if t is the dimensional theory of a mass dimension and a velocity dimension equipped with certain quantities obeying certain equations (as discussed here), then z(t) is the theory of just such a pair of dimensions, but construed as invertible modules over a local ring r.

with an understanding of how an adjunction between doctrines corresponds to the relationship between a “poorer” doctrine and a “richer” one, as discussed here, all this can be summarized as follows: the (groupoid-enriched) functor t |-> z(t) is the left adjoint of the functor that assigns to a coherently locally ringed topos (x,r) the dimensional environment of all invertible modules of
the local ring object r in the topos x.

the contingent implementation of the concept of dimension in the big zariski topos z(t) is a reflection of the basic “functorial” nature of algebraic geometry, how algebraico-geometric objects (such as “the projective plane”) manifest themselves simultaneously over many different commutative rings (in this case local rings). this functoriality is the reason for the word “stack” in the phrase “moduli stack”, a stack being originally a functorially varying groupoid of some sort. the concepts of “moduli stack” and of “classifying topos” are in a sense re-inventions of each other. roughly, whenever toposes appear in algebraic geometry, they’re associated with moduli stacks, and whatever the moduli stack is classifying, the topos classifies essentially the same kind of thing (for example perhaps some kind of lie algebra, to return to this example).

Posted by: james dolan on June 30, 2009 3:45 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Here’s a little puzzle, to steer the conversation in the direction of what Jim was actually explaining.

He gives a recipe for getting a graded commutative algebra from a theory of physics with certain ‘dimensions’, certain ‘quantities’ having these dimensions, and certain ‘relations’ between these quantities.

He also sketches a recipe, known at least in part to algebraic geometers, for getting a projective variety — or perhaps something more general! — from a graded commutative algebra.

Try an easy example. Say we have the theory of two particles each of which has a mass… and that’s all!

What graded commutative algebra does this give, following Jim’s recipe?

And what variety do we get from that?

Posted by: John Baez on June 9, 2009 1:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I haven’t really watched the videos, but you have two numbers (i.e. masses) which make sense only up to scaling (=choice of units). This looks to me like the projective line, which corresponds to the K[x,y]K[x,y], with the usual grading. (Here KK is whatever ring you want your masses to be in.) So that’s my guess.

Posted by: James on June 9, 2009 3:01 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Yes! Your guess is right, though for full credit you need to tell me how you’re making K[x,y]K[x,y] into a graded commutative algebra.

Here’s the idea:

We want to get a graded commutative algebra, say AA, from a theory of physics with certain ‘dimensions’, certain ‘quantities’ having these dimensions, and certain ‘relations’ between these quantities.

Note: when I say ‘graded commutative algebra’ I mean this in a very general sense. We don’t need the grades to be integers; they can live in any abelian group GG. In practice, this will often be a free abelian group.

Here’s how it works:

  • The ‘dimensions’ are the generators of the abelian group GG which serves as the grading group for our algebra.
  • Each ‘quantity’ is a generator of our algebra AA. The ‘dimension’ of that quantity says what grade this generator lives in.
  • Each ‘relation’ between quantities gives us an equation that holds in our algebra AA. We are only allowed to equate quantities that live in the same grade!

You’ll notice this is like a ‘generators and relations’ description of an algebra, except now there are three levels instead of two: dimensions, quantities and relations. There’s a profound reason for that — it has to do with categorification.

But never mind. Let’s do the example.

I said “we have the theory of two particles each of which has a mass… and that’s all!”

So, I’m telling you one dimension, mass. And this generates a free abelian group GG, namely \mathbb{Z}. Its elements are

...,mass 2,mass 1,1,mass,mass 2,... ..., mass^{-2}, mass^{-1}, 1, mass, mass^2, ...

This should make you think of dimensional analysis in physics. And that’s the whole point.

Next, I’m saying there are two quantities with dimensions of mass: the masses of our two particles, say m 1m_1 and m 2m_2.

So, our graded commutative algebra has two generators, m 1m_1 and m 2m_2, both of which live in grade 1.

And then I said “that’s all!” — so there are no relations.

So, the answer is

G=G = \mathbb{Z}

A=K[m 1,m 2]A = K[m_1, m_2]

where both m 1m_1 and m 1m_1 live in grade 1.

And as you know, this graded commutative algebra is related to the projective line. We should say much more about the physical meaning of that process: the process of getting a space from our graded commutative algebra. But this is enough for now.

Posted by: John Baez on June 9, 2009 7:39 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Your guess is right, though for full credit you need to tell me how you’re making K[x,y] into a graded commutative algebra.

he already did! he said “with the usual grading”.

Posted by: james dolan on June 11, 2009 12:07 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

True. But below he sounds unsure as whether we should be using graded rings at all, much less GG-graded rings for various choices of abelian groups GG.

What the heck — I’ll give him full credit.

(Sorry: I’ve been grading finals, which puts me in a supercilious and judgmental mood. But luckily I’m done with that. In fact now I’m in the airport, waiting for a plane to Paris.)

Posted by: John Baez on June 11, 2009 4:03 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Just to be sure, JD is talking completely about graded rings and hence *projective* algebraic geometry, right? While permutation groups are important examples of groups, they’re certainly not the whole story. Similarly, projective algebraic geometry is not by any means the same as algebraic geometry.

Unfortunately I don’t have anything substantial to add. There’s lots of video time and not much written, so it would take a while to go through it all.

Posted by: James on June 9, 2009 2:33 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

The stuff that’s written up so far is actually quite densely packed with information, and all the puzzles I plan to pose can be answered by reading that. No understanding of doctrines, geometric morphisms between topoi, 2-monads, or other complicated crud is required.

The idea of these puzzles is to show how easy it is to understand large hunks of algebraic geometry as the study of ‘dimensional theories’ — things like theories of physics, where the quantities involved have dimensions like

masslength 2/time 2,mass length^2 / time^2,

and it’s against the rules to add quantities with different dimensions.

In my answer to the last puzzle, I explained how to turn dimensional theories into graded commutative algebras.

Here’s the next puzzle. It’s probably way too easy for you, James, so I hope someone else does it. These puzzles are supposed to gradually get more interesting. If they get too interesting too soon nobody will understand them.

Suppose we have a theory of two particles on a line, each of which has a mass and a velocity… and that’s all. What graded commutative algebra do we get?

Extra credit: find a ‘morphism’ between the dimensional theory in this puzzle and the dimensional theory in the last puzzle. Even if you don’t know what a morphism of dimensional theories is, you can still guess, since these theories are related in an obvious way.

Posted by: John Baez on June 9, 2009 7:59 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I can give it a try. We have two dimensions, mass and velocity, so our grading group ought to be just the free group on these two generators, i.e. G=massvelocityG=\mathbb{Z}mass \oplus\mathbb{Z}velocity. The quantities are the masses m 1,m 2m_1, m_2 and velocities v 1,v 2v_1,v_2 of the two particles. Hence A=k[m 1,m 2,v 1,v 2],A=k[m_1,m_2,v_1,v_2], with the grading specified by |m 1|=(1,0)=|m 2|,|v 1|=(0,1)=|v 2|.\vert m_1\vert=(1,0)=\vert m_2\vert,\quad \vert v_1\vert=(0,1)=\vert v_2\vert. In other words, AA is the ring k[m 1,m 2]k[v 1,v 2]k[m_1,m_2]\otimes k[v_1,v_2] with its usual bigraded structure. Geometrically, AA corresponds to 1× 1\mathbb{P}^1\times\mathbb{P}^1. There is a natural inclusion k[m 1,m 2]A,ff1k[m_1,m_2]\rightarrow A, f\mapsto f\otimes 1 which geometrically is the projection of 1× 1\mathbb{P}^1\times\mathbb{P}^1 onto its first factor.

A nice subvariety of 1× 1\mathbb{P}^1\times\mathbb{P}^1 is the curve of bidegree (1,2)(1,2) cut out by the polynomial m 1v 1 2m 2v 2 2m_1v_1^2-m_2v_2^2.

Posted by: Johan Alm on June 10, 2009 9:32 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Excellent!

So, you’re saying that that our new theory (two particles with mass and velocity) is some sort of ‘product’ of two copies of our old theory (two particles with mass).

But technically, I guess you’re saying it’s a kind of coproduct, because you’re hinting that there are two ways to include old theory into our new one.

But when we convert theories into spaces, things get turned around. We see that the space for our old theory is the projective line, and the space for our new theory is a product of two copies of the projective line. That’s because this process of converting theories into spaces is contravariant.

And we should probably talk more about that process right now.

Given a theory TT — or to be precise, a dimensional theory — we start by forming a category hom(T,K)hom(T,K) of ‘models’ of this theory in some rather simple environment KK. Then we form the set of isomorphism classes of models — usually called the ‘moduli space’ of models. And this is almost the space we want.

But what’s KK? KK is the boring theory where all quantities are dimensionless: the theory of dimensionless numbers! As a graded commutative algebra, its grading group is the trivial group, so there’s only one grade. And, the quantities in this grade are just the numbers in our favorite commutative ring kk — the one we fixed right at the start of this game.

In other words, KK is just a glamorized version of the ring kk.

Okay, let’s see if everyone understands this stuff. Let me pose two more puzzles.

For starters, what does a morphism

f:TKf : T \to K

amount to when TT is the theory of two particles with mass? We call such a morphism a model of our theory TT. But what does it amount to, concretely?

(This isn’t hard. Remember that we can think of TT and KK as commutative graded algebras. So, ff is just a homomorphism of commutative graded algebras. So, anyone who knows what all these words means can figure out what it amounts to.)

More interestingly, when do two models

f,g:TKf, g: T \to K

count as isomorphic? For this we need to understand morphisms between models. And for this, we need to realize that we can also think of TT and KK as categories of some sort. Namely, ‘dimensional categories’, as defined in the paper.

From this viewpoint, the models f,g:TKf, g : T \to K are actually functors of some sort… and a morphism between models is a natural transformation of some sort.

So, if you’re a category theorist, you can use this to figure out when two models of TT in KK are isomorphic.

There must also be a way to do this thinking of TT and KK as graded commutative algebras, but that might be harder, or less fun.

But there’s also a way to do this quite intuitively, thinking of TT and KK as physical theories — with KK being a physical theory of a very degenerate sort, suitable only for mathematicians: the theory of nothing at all except numbers!

You see, a model of TT in KK is a way of attaching numerical values to physical quantities. But two such recipes should count as isomorphic if they differ only by ‘change of units’. If you’ve ever thought about dimensional analysis, you can guess when this happens without knowing any category theory!

Of course the fun part is how the formal category-theoretic approach to the puzzle matches the intuitive physical approach.

Posted by: John Baez on June 10, 2009 11:12 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

As a dimensional category, TT should have an abelian group of objects freely generated by one object massmass, and morphisms freely generated (as a symmetric monoidal kk-linear category) by two morphisms m 1,m 2m_1,m_2 from II to massmass. Likewise, KK should have one object II with kk as its endomorphisms.

Then a (symmetric monoidal) functor TKT\to K has to send all the objects to II, and is then determined by the images of m 1m_1 and m 2m_2 in kk, i.e. it assigns numbers to the mass of each particle.

Now a (monoidal) natural transformation between two such ff and gg should be determined by the component α mass:f(mass)g(mass)\alpha_mass : f(mass) \to g(mass), which will be an element of kk. In particular, monoidalness of α\alpha means that α I\alpha_I must equal 1k1\in k. Naturality of α\alpha then says that α massf(m i)=g(m i)α I=g(m i)\alpha_mass \cdot f(m_i) = g(m_i) \cdot \alpha_I = g(m_i) for i=1,2i=1,2. In other words, ff and gg are isomorphic just when there is a constant α massk\alpha_mass\in k such that all the masses assigned by ff are just scaled by α mass\alpha_mass from the masses assigned by gg. In other words, ff and gg are related by a change of units.

Posted by: Mike Shulman on June 11, 2009 2:59 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Excellent, excellent! All the details I left out, you filled in!

So we’re working in the 2-category — dare I say doctrine? — where:

  • The objects are dimensional categories: symmetric monoidal kk-linear categories where all objects xx are line objects, meaning there exists an ‘inverse’ x *x^* with xx *I, x \otimes x^* \cong I , and the symmetry S x,x:xxxxS_{x,x} : x \otimes x \to x \otimes x is the identity morphism.
  • The morphisms are dimensional functors, meaning symmetric monoidal kk-linear functors.
  • The 2-morphisms are dimensional natural transformations, meaning monoidal natural transformations.

And, this 2-category is equivalent to a 2-category where:

  • The objects are graded commutative algebras AA with arbitrary abelian grading groups GG.
  • The morphisms are homomorphisms of graded commutative algebras AAA \to A' which ‘ride atop’ a group homomorphism GGG \to G'.
  • The 2-morphisms are something people don’t seem to talk about in graded commutative algebra land. (Or maybe they do and I just haven’t heard them? In any case, I leave this item as a puzzle.)

Thinking of dimensional categories as graded commutative algebras makes contact with lots of nice math… like, umm, algebraic geometry! But we also want to think and talk like categorical logicians, so we’ll also call dimensional categories dimensional theories. Given such a theory TT, we call the category hom(T,K)hom(T,K) the category of models of TT. Here KK is the rather bland dimensional theory we’ve already been discussing, with one object, and our commutative ring kk as endomorphisms.

Now, back to our example where TT is the ‘the theory of two particles with mass’. Mike worked out when two models are isomorphic:

In other words, ff and gg are isomorphic just when there is a constant α massk\alpha_{mass} \in k such that all the masses assigned by ff are just scaled by α mass\alpha_{mass} from the masses assigned by gg. In other words, ff and gg are related by a change of units.

Right! Ain’t it cool?

(To be pathetically pedantic, of course we need α mass\alpha_{mass} be invertible. Usually I have in mind that kk is a field, in which case we just need α mass\alpha_{mass} to be nonzero.)

Now maybe someone — someone else? — can take advantage of Mike’s hard work and deliver the punchline: what’s the moduli space of models in this case? In other words: what’s the set of isomorphism classes of models?

Posted by: John Baez on June 11, 2009 4:40 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

John asks

what’s the set of isomorphism classes of models?

It’s looking like 1(k)\mathbb{P}^1(k), at least so long as a model can’t choose both particles to be massless. But then why not? So should we add in an extra class for the singleton (0,0)(0, 0)?

Posted by: David Corfield on June 11, 2009 1:18 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

It’s looking like ℙ^1(k), at least so long as a model can’t choose both particles to be massless. But then why not? So should we add in an extra class for the singleton (0,0)?

as mike shulman hinted, algebraic geometers normally rule out this extra degenerate model simply by declaring it (or rather the corresponding graded ideal) “irrelevant”.

it’s a bug/feature in the ointment. there’s more to say about it, of which i’ll only try to say a little bit here.

one way to visualize a projective line is as “the pencil of lines through a point in the plane”. but what do you do with the point itself? being a fixed point of the “renormalization group” of scaling transformations centered at the point makes it a bad/interesting point when you’re modding out by that group. you can throw it away because you don’t like it, but if you keep it then you may find that it acquires automorphisms from being fixed under those scaling transformations. when a point has automorphisms it begins to seem not so point-like and to suggest the usefulness of thinking of them as “models” instead of as “points”.

Posted by: james dolan on June 11, 2009 9:14 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

And yet physically it makes sense to have two massless particles. If you allow one of the particles to have zero mass, why not both?

Then again maybe we shouldn’t be thinking too physically. As kk is a ring we seem to have no problem with ‘negative’ masses.

Posted by: David Corfield on June 12, 2009 9:17 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Then again maybe we shouldn’t be thinking too physically. As kk is a ring we seem to have no problem with ‘negative’ masses.

Well, if kk is a ring. This stuff makes about as much sense if kk is any rig, and most spaces of physical quantities are more modules over R +=[0,[\mathbf{R}^+ = [0,\infty[ than modules over R=],[\mathbf{R} = ]-\infty,\infty[.

In particular, mass as it appears in known physics is a quantity whose values form a module over R +\mathbf{R}^+. Sure, you can talk about negative masses if you want, but even then, there is a distinction between positive and negative mass. Ultimately, we have an equivalence (of dimensional categories) between R +\mathbf{R}^+lines and oriented R\mathbf{R}lines.

On the other hand, some physical dimensions do seem to really be R\mathbf{R}lines; that of electric charge is an example. So perhaps we should work in the category of possibly oriented R\mathbf{R}lines. (Note that these are all line objects; the identity line is oriented, and the square of an unoriented line is oriented. So this is a dimensional category.)

Posted by: Toby Bartels on June 13, 2009 3:13 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

…mass as it appears in known physics is a quantity whose values form a module over R +R^+

On the other hand, some physical dimensions do seem to really be RRlines; that of electric charge is an example. So perhaps we should work in the category of possibly oriented RRlines.

Don’t we need futher rigs around when we do classical mechanics? Action has the dimension of Energy multiplied by Time. As a quantity its values are taken in the rig min\mathbb{R}^min.

Posted by: David Corfield on June 15, 2009 11:13 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Don’t we need futher rigs around when we do classical mechanics? Action has the dimension of Energy multiplied by Time. As a quantity its values are taken in [the rig of real numbers supplemented by positive infinity, with the formal multiplication operation being actual addition and the formal addition operation being actual minimization].

there’s a lot of interesting ideas here. algebraic geometry can certainly be generalized from living over a commutative ring to living over a commutative rig (for example so-called “tropical algebraic geometry”), and in fact it can be generalized a lot farther than that.

also, rig structures that vary as a function of a parameter are important at a foundational level in physics: “(helmholtz) free energy” is always additive with respect to the conjunctive combination of thermodynamic systems, but it obeys a more complicated and temperature-dependent rule with respect to the disjunctive combination of thermodynamic systems. this gives a temperature-dependent rig structure on the set of energy differences (with the confusing aspect that the formal multiplication in the rig is the actual addition of energy differences), and at zero temperature the formal addition of energy differences converges towards minimization. (that is, at zero temperature a system with the option to be in one of two states will surely choose the lesser-energy state; at very low positive temperatures this is only a very good bet.)

combining these ideas together with the foundational role that dimensional analysis plays in both physics and algebraic geometry suggests a lot of interesting possibilities, way too many for me to say much useful about them all at the moment. instead i’ll retreat to discuss a possibility that embraces in a negative way the idea of varying the formal addition operation in a rig: by completely ignoring the concept of addition, thus achieving invariance with respect to variation of it.

as discussed here, the kind of algebraic geometry that “ignores addition” is the study of so-called “toric varieties”. we can define a “toric dimensional theory” as a dimensional theory in which the axioms of the theory (that is, the generating equations between the generating quantities) make no mention of addition.

for example, consider the “segre embedding” of p^1 X p^1 into p^3, as discussed here. addition doesn’t appear in the generating equations for p^1 X p^1 or for p^3, so these dimensional theories are toric dimensional theories. (as a matter of fact no generating equations are needed for these dimensional theories, so it’s easy for addition not to appear in them.) moreover addition doesn’t appear in the formula for the segre embedding itself, so the embedding is a “toric morphism” between toric varieties.

we can formalize the idea of “ignoring addition” more systematically by saying that instead of founding algebraic geometry on the symmetric monoidal category of vector spaces over some field (or of modules over some commutative rig), we’re going to study a version of it founded on the symmetric monoidal category of sets. thus a toric dimensional theory is now defined as a symmetric monoidal category where every object is a line object (meaning invertible and with trivial self-braiding). this is just like the definition of an ordinary dimensional theory except that now the hom-sets in the theory are only required to be sets, instead of being vector spaces.

furthermore, the theorem that says that dimensional theories can be re-interpreted as graded commutative algebras extends to a generality great enough to cover the toric case: a toric dimensional theory can now be re-interpreted as a graded commutative monoid.

(puzzle: in as relatively lowbrow terms as possible, what exactly do we mean by “graded commutative monoid” here??)

to me, it was already surprising when i found out how such a category-theoretically fundamental concept as “dimensional theory” plays such an important role in algebraic geometry; it suggested to me that category theory and algebraic geometry are closely related in ways that aren’t generally well-appreciated. but this feeling became even stronger for me when i found out about the concept of toric dimensional theory, since this concept is category-theoretically even more fundamental: it’s just the concept of “symmetric monoidal category”, supplemented by the requirement that all objects be invertible (and have trivial self-braiding, though, again, it’s interesting to consider relaxing this latter requirement).

there’s still a lot more work to be done understanding in detail how the doctrine of toric dimensional theories (together with related more expressive doctrines) relates to the standard general theory of toric varieties.

Posted by: james dolan on June 22, 2009 7:56 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

John wrote approximately:

Now maybe someone — someone else? — can take advantage of Mike’s hard work and deliver the punchline: what’s the moduli space of models of the theory of two particles, where all the particles have is mass?

David wrote:

It’s looking like 1(k)\mathbb{P}^1(k), at least so long as a model can’t choose both particles to be massless.

Right, exactly!

But then why not? So should we add in an extra class for the singleton (0,0)(0,0)?

Good question! Algebraic geometers explicitly throw out the origin before defining the projective line to consist of equivalence classes of points (m 1,m 2)(m_1,m_2) under rescaling. They call it ‘irrelevant’. But one of the fun things about our way of looking at algebraic geometry is to suggest that maybe it’s not so irrelevant after all.

It’s amusing: projective geometry began life by taking the ordinary line and throwing in a ‘point at infinity’, but now we may want to also throw in the ‘point at zero’.

Regardless of whether this point is ‘irrelevant’, it’s certainly different than the rest. It’s ‘stacky’, meaning it has automorphisms. In other words, it’s mapped to itself by the rescaling transformations

(m 1,m 2)(αm 1,αm 2)(m_1, m_2) \mapsto (\alpha m_1 , \alpha m_2)

So, if we include it, we should treat our ‘moduli space of models’ as something more like a ‘moduli stack’. Which hints yet again that categorification is built into algebraic geometry, right at the point where we pass from affine algebraic geometry to projective algebraic geometry. That makes sense, since this is the point where we start studying a space by looking at the dimensional category of line bundles over it, instead of the algebra of functions on it.

On the other hand, if we really don’t like the ‘irrelevant point’, we can explicitly rule out this model if we want, as soon as we get to a doctrine that’s a bit more powerful than the doctrine of dimensional categories. There’s a lot to say about that, but not today.

Posted by: John Baez on June 12, 2009 12:36 PM | Permalink | Reply to this

WHICH field covers all masses?; Re: Algebraic Geometry for Category Theorists

Okay, I’ll accept that mass needs to be nonzero for this all to work – though that excludes some Physical theories.

But WHICH field covers all masses? And must it really be a field (in the Math sense, not the Maxwell forcefield sense)? I’ve published on complex mass (standard for decaying particles; nonstandard too long to include in this comment). But why can’t a physical object have quaternionic mass? Octonionic? Clifford Algebraic? Grassmanian?

Posted by: Jonathan Vos Post on June 14, 2009 5:50 PM | Permalink | Reply to this

Re: WHICH field covers all masses?; Re: Algebraic Geometry for Category Theorists

Okay, I’ll accept that mass needs to be nonzero for this all to work […]. But WHICH field covers all masses? And must it really be a field […]?

I think that part of Jim's perspective is that we should be able to handle the zero mass as easily as any other; the ‘irrelevant’ point is tricky (since it's not rigid) but can be dealt with. As for using other fields, rings, or even rigs, that fits pretty well into this framework. (And if anything, algebraic geometry is even easier with complex than real scalars.)

Posted by: Toby Bartels on June 14, 2009 6:54 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

john has posed a couple of puzzles of the general form: consider such and such a dimensional theory; what graded commutative algebra does it correspond to?

i have another such puzzle, for people who haven’t seen it before. but while the theories that john has described have been physics-motivated, the theory in this next puzzle is of a different flavor, designed to illustrate a different aspect of the doctrine of dimensional theories.

consider the dimensional theory of “a pair (d,e) of lines equipped with a lie algebra structure on the direct sum d+e”. what graded commutative algebra does it correspond to?

(your description of the graded commutative algebra should be fairly explicit, as usual; for example by explicit generators and relations. extra credit for stuff like giving the dimension of the moduli space of models of the theory, or for generalizing to the case of an n-tuple of lines where n>2, for example n=3.)

Posted by: james dolan on June 13, 2009 8:46 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Spoiler warning: this puzzle is worked out by James D in one of the videos (the third one, if memory serves).

Posted by: Todd Trimble on June 13, 2009 8:58 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Maybe I’ll try doing a bit of this puzzle. We’ve got a dimensional theory with two lines in it, say dd and ee. So, in particular, we have a category where all the objects are tensor products of copies of dd and ee and their inverses. So, all the objects look like this:

d me n d^{\otimes m} \otimes e^{\otimes n}

where m,nm,n \in \mathbb{Z}.

So, the objects form an abelian group under tensor product, and this abelian group is 2\mathbb{Z}^2. That’s the ‘grading group’ for our dimensional theory, if we choose to think of it as a commutative graded algebra.

But what about the morphisms? Jim wants us to think of ded \oplus e as equipped with the structure of a Lie algebra, meaning we have a morphism called the ‘Lie bracket’:

[,]:(de)(de)de[\cdot,\cdot] : (d \oplus e) \otimes (d \oplus e) \to d \oplus e

satisfying the usual Lie algebra axioms.

At this point you may protest: but our category doesn’t have direct sums, just tensor products!

In fact, that may be why so many of you refused to tackle this puzzle.

Okay, but the point is that if we did have direct sums, we could reexpress the Lie bracket morphism in terms of various pieces, like

[,]:ddd[\cdot,\cdot] : d \otimes d \to d

and

[,]:dde[\cdot,\cdot] : d \otimes d \to e

and

[,]:ded[\cdot,\cdot] : d \otimes e \to d

and so on. And, we can formulate the Lie algebra axioms — antisymmetry, Jacobi identity — in terms of these various pieces. We get a bunch of equations between morphisms in our dimensional theory.

Now consider a model of this theory. It’s just a 2-dimensional Lie algebra broken up as a direct sum of 1-dimensional spaces: ded \oplus e.

What can such a model be like?

Posted by: John Baez on June 17, 2009 9:58 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Just to be sure, JD is talking completely about graded rings and hence *projective* algebraic geometry, right?

i’d have to know how narrowly you’re construing “projective algebraic geometry” to decide how much i’d agree with your “hence” here. allen knutson gave some hints above about how graded rings go beyond the narrowest interpretation of “projective”, and hopefully those ideas will get expanded upon some more in this thread.

but i’m not talking just about graded rings. when i talk about the doctrine of dimensional theories i’m basically talking about graded rings, but i’m also talking about other doctrines. in particular in one of the doctrines that i’m talking about, the theories are like categories of coherent sheaves, so the geometric objects that i’m talking about include any geometric object that can be recovered from the category of coherent sheaves over it; in other words, any geometric object to which a “gabriel-rosenberg theorem” applies. i’m not sure exactly how general such geometric objects are.

While permutation groups are important examples of groups, they’re certainly not the whole story.

hmm, there may be some sort of pattern here- from my point of view, which includes not defining “permutation group” in an overly narrow way, permutation groups _are_ the whole story. i guess that i like stories unified by a simple pervasive theme.

Similarly, projective algebraic geometry is not by any means the same as algebraic geometry.

my impression of what algebraic geometers are interested in is that projective algebraic geometry is a _very_ large part of it, their “bread and butter”.

as i said in the lectures, the role of the doctrine of dimensional theories (aka graded commutative rings) is, roughly, to correspond to an older tradition in algebraic geometry, according to which a variety is automatically assumed to come equipped with a specific projective embedding.

(when i saw this older tradition discussed it was not actually referred to as a “tradition” but rather as a “disease”. it seems clear that it was a tradition though.)

the role of the higher doctrines that i’m talking about is, roughly, to correspond to a more modern tradition of studying a variety or scheme independent of any projective embedding.

Posted by: james dolan on June 11, 2009 11:29 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Yeah, soon after I posted that John said something about other doctrines and coherent sheaves. It will be interesting to read!

Regarding the permutation group analogy, I don’t think we really disagree. I just meant to say that a projective embedding of a variety is an auxiliary piece of data, much like giving an action of a group on some set. These are both useful tools for studying the original variety / group, but the variety / group exists prior to the embedding / action. (Of course, groups do have natural permutation representations, the regular representations, whereas varieties essentially never have a distinguished projective embedding. So the analogy is pretty poor, but I couldn’t think of anything better.)

Posted by: James on June 12, 2009 12:46 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Of course, groups do have natural permutation representations, the regular representations, whereas varieties essentially never have a distinguished projective embedding.

hmm, i think that this raises some interesting questions that i hope i can formulate precisely enough to ask here relatively soon.

Posted by: james dolan on June 12, 2009 5:07 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Of course, groups do have natural permutation representations, the regular representations, whereas varieties essentially never have a distinguished projective embedding.

you can however canonically construct a dimensional theory from a variety:

take the group of dimensions to be the group of divisors on x, and given divisors d1,d2 take hom(d1,d2) to be the vector space of meromorphic functions on x whose principal divisor is d2-d1. composition of morphisms is given by multiplication of functions and the rest of the structure of the dimensional theory should be clear.

some questions:

what version of the divisor concept “should” we use here?

in what generality (what class of varieties, or perhaps something more general than varieties) does this construction succeed in giving a dimensional theory?

is the classification of the models of the resulting dimensional theory “known”? particularly the non-rigid models, those possessing a non-trivial automorphism.)

in what generality is there a natural bijection from the points of the variety to the isomorphism classes of rigid models?

part of the point here is that dimensional theories have to do with not just projective embeddings, but multi-projective embeddings, and some sort of infinite limits of multiprojective embeddings. varieties may not have distinguished projective embeddings, but they _do_ have distinguished embeddings of a more general type. not too surprising, since you can just take _all_ the projective (near-)embeddings of a variety and combine them together, which is essentially what the construction that i described above is doing.

in certain situations there are concepts of theories of a doctrine being “morita-saturated”. the somewhat gigantic dimensional theories arising from the above construction are “morita-saturated” in a certain sense. i hope that i get around to explaining more about this concept of “morita-saturation” (and related concepts such as “morita equivalence” of theories) and how it relates to stuff that we’re doing here.

Posted by: james dolan on June 13, 2009 5:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I’m still not really sure what a lot of the words you’re using mean, but I would guess that what you want is algebraic equivalence of divisors, which would make the group of dimensions the Neron-Severi group. Then hopefully, by the Neron-Severi theorem, the associated multi-projective variety would actually be a finite product of projective spaces. And then perhaps the induced map to the multi-projective space is an embedding if (and only if) the original variety admits a projective embedding? This would be nice (and would in spirit prove false what I said about about distinguished embeddings, though there are still varieties admitting no projective emdeddings). I don’t know if this is well-known. (Despite some rumors to the contrary, I’m not really an algebraic geometer, and definitely not a projective geometer.)

But since don’t really know what a lot of the words you’re using mean, I think it would be instructive to look at the projective line. Then the divisor group is ZZ, the group of integers under addition, and Hom(m,n)\mathrm{Hom}(m,n) is presumably the vector space of homogeneous polynomials of degree nmn-m in two variables x,yx,y (ie the space of maps between the corresponding line bundles). Apparently this should induce a map from the projective line to a variety which should be thought of as multi-projective. It seems reasonable to bet that it’s the identity map. Does anyone feel like writing out this example in detail?

Posted by: James on June 13, 2009 7:46 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I would guess that what you want is algebraic equivalence of divisors, which would make the group of dimensions the Neron-Severi group.

i think that the most relevant equivalence relation on divisors here is “linear equivalence”, since that’s the one that’s supposed to correspond to isomorphism of line bundles (and line bundles are basically the same thing as “dimensions”).

(i’d really like to understand more about the different equivalence relations on divisors that are used, though, as discussed a bit here for example.)

so i don’t think that we’re getting something as strong as a canonical embedding into a finite product of projective spaces in most cases (though it depends to some extent on how you interpret “canonical” and “most”).

anyway, my main point was that there’s a certain tradeoff with dimensional theories: you can get a dimensional theory that corresponds to a given projective variety x without showing bias towards any particular projective embedding of x, but only at the expense of including (in general) a relatively huge number of independent dimensions in the theory. (and that this is somewhat typical of situations where you’re dealing with some kind of “morita equivalence” of theories.)

But since don’t really know what a lot of the words you’re using mean, I think it would be instructive to look at the projective line. Then the divisor group is Z, the group of integers under addition, and Hom(m,n) is presumably the vector space of homogeneous polynomials of degree n−m in two variables x,y (ie the space of maps between the corresponding line bundles). Apparently this should induce a map from the projective line to a variety which should be thought of as multi-projective. It seems reasonable to bet that it’s the identity map. Does anyone feel like writing out this example in detail?

i should definitely try to work this example out in fairly great detail. i’m not feeling quite up to it yet though.

Posted by: james dolan on June 14, 2009 10:10 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

If you want linear equivalence, then as you say, you probably won’t be able to have any finiteness statements, though maybe you don’t care about that. For instance, the group of divisors up to linear equivalence on an elliptic curve over the complex numbers is uncountable.

Posted by: James on June 15, 2009 12:31 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I wouldn’t take the “disease” comment as very indicative of algebraic geometers’ views. In modern terminology, giving a specific projective embedding (modulo automorphisms of the ambient projective space) is the same as equipping the variety X with a very ample line bundle (the pull-back of the line bundle O(1), whose global sections are linear homogeneous polynomials, on the projective space). Such a line bundle is often referred to as a polarization of X.

(I think that “very ample” is chosen just to indicate that such a line bundle has a “very ample” supply of global sections, while the “polarization” terminology, which may seem a bit opaque, is used for historical reasons: it dates back to Riemann, as far as I know.)

Algebraic geometers frequently study varieties equipped with polarizations. (For example, polarizations often have to be included if one wants to get reasonable moduli spaces.)

Posted by: Matthew Emerton on June 15, 2009 5:32 AM | Permalink | Reply to this

“Line” Stuff

Question from the peanut gallery…

If you could start over and give “line bundle” and consequently “line object” any name you wanted, what would you call them?

Posted by: Eric on June 11, 2009 5:24 AM | Permalink | Reply to this

Re: “Line” Stuff

If you could start over and give “line bundle” and consequently “line object” any name you wanted, what would you call them?

hmm, i’m not sure that i can answer this question- it doesn’t seem counterfactual enough.

coincidentally, though, there’s a school of thought that says that you shouldn’t say “x bundle” (or “bundle of x’s”) but rather “x”, and there’s another school of thought (or maybe it’s the same school of thought?) that says that you shouldn’t say “x object” but rather “x”. which seem to converge at calling them “lines”, which doesn’t seem bad.

i like “dimension” too, though.

Posted by: james dolan on June 11, 2009 8:44 AM | Permalink | Reply to this

Re: “Line” Stuff

The thing that sparked my question was the definition:

Definition. An object xx in a symmetric monoidal category KK is called a line object if it has an inverse object with respect to tensor product, and if the canonical “switching” morphism xxxxx\otimes x\to x\otimes x is the identity morphism.

I’m not sure what it is about this definition that warrants the word “line”. I’m guessing that a better name for this would also serve as a better name for “line bundle”. I know the terminology will never change, but a better descriptive name would probably help me overcome some mental barriers.

Posted by: Eric on June 11, 2009 2:56 PM | Permalink | Reply to this

Re: “Line” Stuff

The thing that sparked my question was the definition:

Definition. An object x in a symmetric monoidal category K is called a line object if it has an inverse object with respect to tensor product, and if the canonical “switching” morphism x⊗x→x⊗x is the identity morphism.

I’m not sure what it is about this definition that warrants the word “line”. I’m guessing that a better name for this would also serve as a better name for “line bundle”. I know the terminology will never change, but a better descriptive name would probably help me overcome some mental barriers.

oh. no, i don’t think that we need a better descriptive name here; we probably just need a better explanation of what about the definition warrants the word “line”. (i may have completely neglected to give such an explanation.)

the intuition behind it is pretty rudimentary: multi-dimensional objects have no tensor inverses because the multiplicativity of tensor product with respect to dimensionality would require their inverses to be fractional-dimensional; thus having a tensor inverse is the hallmark of a one-dimensional object (aka a “line”, aka a “dimension”).

the other clause in the definition of “line object”, requiring that its “self-braiding” be trivial, seems less crucial to me, so it should be fun to experiment with omitting that clause.

Posted by: james dolan on June 11, 2009 10:40 PM | Permalink | Reply to this

Re: “Line” Stuff

A standard name for an invertible object is “a unit”.

Which is maybe noteworthy (maybe not), in view of your interpretation of these objects in the context of physical units.

Posted by: Urs Schreiber on June 11, 2009 10:55 PM | Permalink | Reply to this

Re: “Line” Stuff

It is indeed noteworthy!

There’s a funny opportunity for confusion here. All the objects xx in a dimensionless category are invertible — these are our dimensions. But not all morphisms f:1xf: 1 \to x are invertible — these are our ‘quantities of dimension xx’. But if ff is invertible, we can treat it as a ‘unit of measurement’ for quantities of dimension xx. That is, it lets us turn any quantity of dimension xx into a dimensionless quantity.

In a while we should do some exercises where we take one of the simple theories I’ve been discussing and modify it by ‘working in a system of units where we treat mass as dimensionless’. This amounts to throwing in an invertible morphism f:1massf : 1 \to mass.

Posted by: John Baez on June 12, 2009 2:13 PM | Permalink | Reply to this

Re: “Line” Stuff

A standard name for an invertible object is “a unit”.

Historically, I'm pretty sure that this coincidence is no coincidence (to use the word ‘coincidence’ in two different senses, although these are of course also historically related).

In general, we can define a unit to be any value uu such that, given any value xx, there exists a unique scalar kk such that x=kux = k u. Of course, what this means depends on what we mean by ‘value’ and ‘scalar’ (as well as the operations that appear in the equation).

  1. So if a value is a physical length and a scalar is a positive real number, then a unit is precisely (at least at the level of approximation that one usually does physics) a unit of length.

  2. Or let KK be a field and VV be a vector space over KK; if a value is an element of VV and a scalar is an element of KK, then a unit exists if and only if the dimension of VV is 11, in which case a unit is precisely a nonzero element of VV.

  3. Or let RR be a commutative ring; if a value and a scalar are each an element of RR, then a unit exists if and only if the ring is unitary (as category theorists tend to assume it is), in which case a unit is precisely an invertible element of RR.

  4. Or let CC be a symmetric monoidal category; if a value and a scalar are each an object of CC, then a unit exists (as the identity, or unit, object), and every line object is a unit. (There may be other units besides the line objects; since this is a categorified case, however, perhaps we should include a coherence condition that will force every unit to be a line object.)

Now let KK be a field again, and let CC be the symmetric monoidal category of vector spaces over KK. Then it is precisely the units (in sense 4) that have units (in sense 2); this is an example of the microcosm principle.

Posted by: Toby Bartels on June 13, 2009 2:34 AM | Permalink | Reply to this

Re: “Line” Stuff

Related to this remark, one might note that in algebraic geometry, the standard terminology for the sheaf of sections of a line bundle is “invertible sheaf”.

This is chosen to emphasize that these are precisely the sheafs (on the given variety) that have an inverse (up to isomorphism) under the tensor product operation on sheaves.

Posted by: Matthew Emerton on June 15, 2009 5:15 AM | Permalink | Reply to this

Re: “Line” Stuff

Eric wrote:

I’m not sure what it is about this definition that warrants the word “line”. I’m guessing that a better name for this would also serve as a better name for “line bundle”. I know the terminology will never change…

… because Jim invented it and he’s incredibly stubborn?

but a better descriptive name would probably help me overcome some mental barriers.

Feel free to call it a ‘line bundle’ if you like. As Jim mentioned in his reply to your first question, there’s a certain sophisticated crowd who ruthlessly downplays the distinction between an ‘XX’ and an ‘XX bundle’. The reason is that both of these can be thought of as an ‘XX object’, thanks to internalization. And then after a while you get sick of saying ‘XX object’ and just say ‘XX’!

For example, I used to use ‘torsor’ as the name for a fiber of a principal bundle, and I got really confused when that sophisticated crowd used ‘torsor’ to mean a principal bundle. I would have been happy if they said ‘torsor bundle’. But then I grew up: I learned that a torsor bundle really is a ‘torsor object’ in some category of bundles, and then I got sick of saying ‘torsor object’ and acquiesced to simply saying ‘torsor’.

But it’s not good to pretend you’re sophisticated before you actually are. So, you should feel perfectly fine about calling the objects in a dimensional category ‘line bundle objects’ or ‘line bundles’ or ‘line objects’ or ‘lines’ — whatever aids your intuition most.

Posted by: John Baez on June 12, 2009 8:57 AM | Permalink | Reply to this

Re: “Line” Stuff

When I first saw this definition it looked like rubbish, since the switch map in a symmetric monoidal category is almost never the identity. In fact, that’s basically what prevents every symmetric monoidal category from being equivalent to a strictly symmetric one. But then when I looked again at the example of line bundles it made sense: the switch map for the unit object (by which I mean the identity object of the monoidal structure) is always the identity, and a line bundle is basically a “twisted” version of the unit object (the ground ring). So it seems to me that a “line object” in general can be thought of as a “twisted form of the unit object.”

Posted by: Mike Shulman on June 12, 2009 5:55 AM | Permalink | Reply to this

Re: “Line” Stuff

When I first saw this definition it looked like rubbish, since the switch map in a symmetric monoidal category is almost never the identity. In fact, that’s basically what prevents every symmetric monoidal category from being equivalent to a strictly symmetric one. But then when I looked again at the example of line bundles it made sense: the switch map for the unit object (by which I mean the identity object of the monoidal structure) is always the identity, and a line bundle is basically a “twisted” version of the unit object (the ground ring). So it seems to me that a “line object” in general can be thought of as a “twisted form of the unit object.”

presumably this is related somehow to the purely homotopy-theoretic theory of k-line bundles for k a topological (commutative) field, which is controlled by cohomology with coefficients in the strict topological abelian group given by the multiplicative group of k, and for that reason exhibits strict symmetry of the tensor product.

Posted by: james dolan on June 25, 2009 1:26 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Will all the wisdom contained in dimensional analysis find its place in this work?

Posted by: David Corfield on June 11, 2009 8:18 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Posted by: David Corfield on June 11, 2009 8:48 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David wrote:

Will all the wisdom contained in dimensional analysis find its place in this work?

Yes.

I mean, that’s one of my personal goals.

I’ll give a little example pretty soon. I’d also like to do a lot more.

Once I read a book that used dimensional analysis to solve (or at least ‘analyze’) tons of practical physics problems. I forget which book that was. I’d be happy if someone could tell me a promising guess. Online material would be even better!

I’d like to take some examples like that, write down the corresponding dimensional theories, work out their moduli spaces of models (just as we’ve been doing here in some much simpler examples), and see if the sometimes surprising power of dimensional analysis looks any different from this viewpoint.

Posted by: John Baez on June 12, 2009 12:54 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Let’s have a go. Perhaps the second of these:

I see Hassler Whitney wrote something on the subject:

  • The mathematics of physical quantities. Part I. Mathematical models for measurement, Amer. Math. Monthly 75 (1968), 115-138.
  • The mathematics of physical quantities. Part II. Quantity structures and dimensional analysis, Amer. Math. Monthly 75 (1968), 227-256.
Posted by: David Corfield on June 12, 2009 4:19 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I look forward to watching this. I’m not sure if I pass the prerequisites, but I will try to hang tight.

Posted by: Bruce Bartlett on June 11, 2009 12:30 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Welcome to my world :)

Posted by: Eric on June 11, 2009 2:53 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Okay, here’s another little puzzle that builds on the ones we’ve done so far.

We started with a dimensional theory TT called ‘the theory of two particles, each with a mass’. The corresponding graded commutative algebra AA is \mathbb{Z}-graded, and elements of degree dd correspond to quantities of dimension mass dmass^d. The graded commutative algebra AA is free on two generators in degree 1, namely the particles’ masses:

A=k[m 1,m 2]A = k[m_1, m_2]

Because the algebra AA is \mathbb{Z}-graded and generated by elements of degree 1, the moduli space of models of the theory TT is naturally a projective variety — at least after we discard the ‘irrelevant model’ where both particles have mass 0. And as we’ve seen, this projective variety is just the projective line k 1k\mathbb{P}^1.

Then we looked at a theory we could call T+TT + T. We actually called it ‘the theory of two particles on a line, each with a mass and velocity’. The corresponding graded commutative algebra is AAA \otimes A, which we think of as \mathbb{Z} \oplus \mathbb{Z}-graded. Elements of degree (d,e)(d,e) correspond to quantities of dimension mass dvelocity e mass^d velocity^e. AAA \otimes A is freely generated by two quantities of degree (1,0)(1,0), namely the particles’ masses, and two quantities of degree (0,1)(0,1), namely their velocities:

AA=k[m 1,m 2]k[v 1,v 2]A \otimes A = k[m_1, m_2] \otimes k[v_1, v_2]

Because the algebra AAA \otimes A is \mathbb{Z} \oplus \mathbb{Z}-graded and generated by elements of degree (1,0)(1,0) and (0,1)(0,1), the moduli space of models of the theory T+TT + T is a ‘biprojective variety’ — at least after we discard some ‘irrelevant models’. And as we’ve seen, this biprojective variety is just k 1×k 1k\mathbb{P}^1 \times k\mathbb{P}^1.

I should admit: I’ve never actually seen the phrase ‘biprojective variety’. What people usually talk about are ‘multiprojective varieties’. These are varieties that live in a product of projective spaces. Just as you can get projective varieties from (sufficiently nice) graded commutative algebras where the grading group is \mathbb{Z}, you can get multiprojective varieties from (sufficiently nice) graded commutative algebras where the grading group is n\mathbb{Z}^n. And right now we’re doing n=2n = 2.

Okay, now for the puzzle. Suppose we take our theory T+TT + T and look at the ‘subtheory’ TT' consisting only of quantities that have dimensions that are powers of momentum:

momentum d=mass dvelocity d momentum^d = mass^d velocity^d

for some dd \in \mathbb{Z}.

This subtheory TT' gives a subalgebra AAAA' \subset A \otimes A. And we can think of this subalgebra as \mathbb{Z}-graded instead of \mathbb{Z} \oplus \mathbb{Z}-graded, since everything in this subalgebra has dimension momentum dmomentum^d for some dd \in \mathbb{Z}.

And the puzzle is: what’s the moduli space of models of the subtheory TT', and how is it related to the moduli space of models of T+TT + T?

This is a pretty open-ended puzzle. If algebraic geometry is your cup of tea, you may know some buzzwords that describe what’s going on here. If category theory is your cup of tea, you may want to formalize what I’m doing a bit more.

And here’s a hint: you might guess that we don’t lose much by going from T+TT + T to the subtheory TT', because there are four obvious quantities with dimensions of momentum:

m 1v 1,m 2v 1,m 1v 2,m 2v 2 m_1 v_1, m_2 v_1, m_1 v_2 , m_2 v_2

and we started out with four quantities in the first place: m 1,m 2,v 1,v 2m_1, m_2, v_1, v_2.

Posted by: John Baez on June 12, 2009 1:57 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I believe the moduli space for TT' will be the image of the Segre embedding of the two copies of the moduli space for TT.

In our case, the Segre embedding takes the “biprojective variety” 1× 1\mathbb{P}^1 \times \mathbb{P}^1 and maps it into 3\mathbb{P}^3 by

(1)((m 1:m 2),(v 1:v 2))(m 1v 1:m 1v 2:m 2v 1:m 2v 2).((m_1: m_2), (v_1: v_2)) \mapsto (m_1v_1 : m_1v_2 : m_2v_1 : m_2v_2).

The image of this map is precisely what we’re looking for. The map is a closed immersion, whose image is cut out by the kernel of the corresponding map of algebras,

(2)k[x 0,x 1,x 2,x 3]k[m 1,m 2]k[v 1,v 2] k[x_0, x_1, x_2, x_3] \to k[m_1, m_2] \otimes k[v_1, v_2]
(3)x 0m 1v 1x_0 \mapsto m_1 \otimes v_1
(4)x 1m 1v 2x_1 \mapsto m_1 \otimes v_2
(5)x 2m 2v 1x_2 \mapsto m_2 \otimes v_1
(6)x 3m 2v 2.x_3 \mapsto m_2 \otimes v_2.

This kernel is generated by the determinant x 0x 3x 1x 2x_0x_3 - x_1x_2, so we conclude that Ak[x 0,x 1,x 2,x 3]/(x 0x 3x 1x 2)A' \cong k[x_0, x_1, x_2, x_3] / (x_0x_3 - x_1x_2), which maps into AAA \otimes A by the above map.

As James mentioned, the fact that this map is an “embedding” in the sense of algebraic geometry (i.e., a closed immersion) cannot be formulated in the context of dimensional theories. To formulate this notion, we need to be able to talk about coherent sheaves on our spaces.

Posted by: Evan Jenkins on June 12, 2009 5:08 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Evan wrote:

I believe the moduli space for TT&#8242; will be the image of the Segré embedding of the two copies of the moduli space for TT.

Right! That’s exactly the buzzword I was alluding to: ‘Segré embedding’. And even better, the obvious ‘inclusion’ morphism of dimensional theories

TT+TT' \to T + T

induces a map of models

hom(T+T,K)hom(T,K) hom(T + T, K) \to hom(T', K)

which gives the Segré embedding

1× 1 3 \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3

when we take isomorphism classes of models and discard the so-called ‘irrelevant’ ones.

And I’m so bad at algebraic geometry that this really helps me feel I understand the Segré embedding!

As James mentioned…

Just to be clear, we’ve got an algebraic geometer named James contributing to this thread, and also a guy who calls himself james dolan, who is the one you’re talking about. I’m calling the former James and the latter Jim.

If Jim Stasheff joins in, we’re doomed.

Posted by: John Baez on June 12, 2009 7:11 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Right! That’s exactly the buzzword I was alluding to: ‘Segré embedding’.

so presumably one reason that the (generalized?) segre embedding is considered noteworthy is that it shows in some (maybe all?) circumstances how to reduce multi-projective varieties to projective varieties.

Posted by: james dolan on June 12, 2009 7:41 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

As you guessed, one key reason that the Segre embedding is important is that it implies that the product of two projective varieties (which is a priori a multi-projective variety, in the terminology you are using) is in fact again a projective variety.

The category of projective varieties is an important subcategory of the category of all varieties (since projective varieties have various properties not shared by all varieties). So the fact that it is closed under products is useful.

Posted by: Matthew Emerton on June 15, 2009 5:09 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

As you guessed, one key reason that the Segre embedding is important is that it implies that the product of two projective varieties (which is a priori a multi-projective variety, in the terminology you are using) is in fact again a projective variety.

The category of projective varieties is an important subcategory of the category of all varieties (since projective varieties have various properties not shared by all varieties). So the fact that it is closed under products is useful.

i hadn’t even thought of that; that’s a good point too. i was thinking more along the lines of how there are varieties such as flag varieties that when you run into them for the first time are (arguably) more naturally thought of as multi-projective varieties than as projective varieties.

most of the thinking that i’ve done about flag varieties was before i started thinking about multi-projective varieties as moduli stacks of dimensional theories; at some point i should try to post here about flag varieties from this (for me) new viewpoint.

Posted by: james dolan on June 16, 2009 8:57 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Since Jim’s puzzle seems to have thrown all the super-geniuses into a silent fit of frantic calculation, here’s one for the rest of us.

We’ve been talking about a theory TT called ‘the theory of two particles, each with a mass’. It’s really a dimensional category with objects called dimensions

...,mass 1,1,mass,mass 2,mass 3,......, mass^{-1}, 1, mass, mass^2, mass^3, ...

and morphisms called quantities, freely generated by these two:

m 1:1mass m_1 : 1 \to mass m 2:1mass m_2 : 1 \to mass

But we can equivalently think of this category as a graded commutative algebra, say AA. This algebra is \mathbb{Z}-graded, where elements of degree dd correspond to quantities of dimension mass dmass^d. It’s free on two generators in degree 1, namely the particles’ masses:

A=k[m 1,m 2]A = k[m_1, m_2]

A model of TT is just a consistent way of assigning numerical values to the masses m 1m_1 and m 2m_2. Since our theory has no equations, ‘consistency’ is automatic in this case. And since there’s no fixed unit of mass around, rescaling these masses gives an isomorphic model. So, the set of isomorphism classes of models — the moduli space of models — is k 1k\mathbb{P}^1, at least after we discard the so-called ‘irrelevant model’ where both particles have mass 0.

But now let’s take TT and change it a bit by assuming we do have a unit of mass!

In other words, let’s take our category and freely throw in an extra quantity

M:1mass M : 1 \to mass

which however we assume to be an isomorphism. This lets us take any quantity of dimension massmass, say

Q:1massQ : 1 \to mass

and turn it into a quantity of dimension 11, namely

M 1Q:11 M^{-1} Q : 1 \to 1

A quantity of dimension 1 is what physicists call a ‘dimensionless quantity’. At least in the situation at hand, it’s just a number in our field kk.

Having thrown a unit of mass into our theory TT, we get a new theory, say T 0T_0.

Now: what commutative graded algebra A 0A_0 does this correspond to? And what moduli space of models do we get?

For extra credit, interpret your result using the language of physics.

Posted by: John Baez on June 15, 2009 2:10 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Gee, this puzzle was supposed to be pretty easy. Maybe people are scared because they don’t see how take a dimensional theory, throw in an isomorphism, and get a new dimensional theory? Maybe you’re scared because you’re not sure what the grading group of this theory is? Maybe you can’t guess what its commutative graded algebra is?

I’m not sure. Tell me what’s wrong, folks!

I think a robust common-sense approach based on dimensional analysis should be enough to solve the puzzle. We’ve got a theory of two particles; all each one has is a mass, and now (for the first time) we have also have a unit of mass in our theory — say, a gram. What is a model of this theory like?

Posted by: John Baez on June 17, 2009 9:42 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Gee, this puzzle was supposed to be pretty easy. Maybe people are scared because they don’t see how take a dimensional theory, throw in an isomorphism, and get a new dimensional theory? Maybe you’re scared because you’re not sure what the grading group of this theory is? Maybe you can’t guess what its commutative graded algebra is?

well i can tell you why i’m a bit unsure as to exactly what answers to your questions you’re looking for. my reasons might be a bit different from some other people’s reasons, at least on the surface, but they’re probably secretly the same- you also seem to be hinting at the same reasons yourself.

for me, the confusion is this:

we’re talking here about graded commutative algebras, but we’ve alluded to some sort of theorem that says that they’re equivalent in some sense to “dimensional categories”. but dimensional categories, being categories, can be distinguished either up to strict isomorphism (often a bad idea), or up to a coarser relation given by some kind of “equivalence of categories” (often a good idea; perhaps here it should be some sort of “symmetric monoidal equivalence” because of dimensional categories being a kind of symmetric monoidal categories.) so this makes me wonder whether there’s also some sort of “coarse equivalence of graded commutative algebras” that we should be concerned with. or more precisely, it makes me try to remember what i concluded about this issue the last time that i thought about it. because it seems like i might need to resolve this issue in order to give sensible answers to your questions.

anyway, i made no attempt to avoid a blatant category-theory bias in the answer that i just gave. maybe someone else should try to give more of an intuitive “dimensional analysis” take on the situation. it has something to do with: when do two dimensions count as “equivalent”, and how do you deal with that? do you just get rid of one of them as being “redundant”, and if so then how do you know which one is the redundant one?? and stuff like that…

Posted by: james dolan on June 17, 2009 10:18 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Jim wrote:

well i can tell you why i’m a bit unsure as to exactly what answers to your questions you’re looking for. my reasons might be a bit different from some other people’s reasons, at least on the surface, but they’re probably secretly the same- you also seem to be hinting at the same reasons yourself.

Yes, I was trying to hint at the technical subtleties of this puzzle without using any technical words to describe them, because I figured those words would make some people more scared to solve the puzzle, rather than less.

And that was probably the right strategy, since with my encouragement David went ahead and solved it.

But there’s still a lot more interesting stuff to say about it…

we’re talking here about graded commutative algebras, but we’ve alluded to some sort of theorem that says that they’re equivalent in some sense to “dimensional categories”. but dimensional categories, being categories, can be distinguished either up to strict isomorphism (often a bad idea), or up to a coarser relation given by some kind of “equivalence of categories” (often a good idea; perhaps here it should be some sort of “symmetric monoidal equivalence” because of dimensional categories being a kind of symmetric monoidal categories.) so this makes me wonder whether there’s also some sort of “coarse equivalence of graded commutative algebras” that we should be concerned with.

Indeed my puzzle was designed to lead us into precisely these deeper waters.

Let me try to say it very non-technically, for all the kids out there who read the nn-Category Café but haven’t yet mastered their symmetric monoidal categories.

In my earlier puzzles, we had physical theories in which the quantities could have various ‘dimensions’, like mass dmass^d or later mass dvelocity emass^d velocity^e. These dimensions formed a set. Since we could multiply and divide them, they actually formed an abelian group. But never mind that: they certainly formed a set, so we felt quite happy about talking about whether dimensions were equal or not.

Now however we have a theory where we have a set of dimensions like mass dmass^d that are not equal to each other — but still they’re all isomorphic to each other.

In other words: having chosen a unit of mass, we can freely convert quantities of dimension mass dmass^d to dimensionless quantities. It’s not that a quantity of dimension mass dmass^d is dimensionless; rather, we can think of it as such.

(When people say “we can think of this as that”, it means they’re treating these things as isomorphic. The “thinking of” process is the isomorphism.)

By the way, we do this sort of thing a lot in physics. For example, in relativity we work in units where the speed of light is 1, which amounts to choosing a unit of velocity and using it to freely convert quantities of dimension velocity dvelocity^d to dimensionless quantities.

When we do this in physics, we usually say ‘set c=1c = 1’, which means ‘pretend the isomorphism between velocities and dimensionless quantities is the identity morphism’.

But, now we’re being trying to be a bit more formal about it, and the difference between equality and isomorphism becomes worth preserving. But it leads to further subtleties! On the one hand, we can imagine a category where I just made mass dimensionless:

mass=1mass = 1

but instead I chose a less brutal approach, and cooked up a category where these dimensions were isomorphic:

Q:1mass Q : 1 \to mass

So, we’ve got two different categories here. These categories are not ‘the same’ — not equal, and not even isomorphic! — but they’re still ‘the same in a way’: we say they’re equivalent.

So this is a place where our treatment of algebraic geometry is perhaps a bit more ‘categorified’ than usual. We’ve said that dimensional theories are categories of a certain sort, but we’ve also said they’re just another way of thinking about commutative graded algebras. However, people don’t seem to talk much about equivalence of commutative graded algebras, while it’s very natural to do so for categories. So, this new viewpoint leads to some fun new things to do.

Luckily, equivalent dimensional theories have the same — or maybe I should say ‘the same’ — moduli stack of models.

So, there are various ways to think about the latest puzzle, but they should all give the same moduli stack of models. And as David saw, this particular moduli stack is really just a moduli space — nothing fancy this time.

Posted by: John Baez on June 17, 2009 4:50 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Maybe people are scared…

…or too busy.

Let’s have a stab while I have a moment. You’d think that having the unit of mass would be as though there were a third (massive) particle, kept in the vaults of a scientific institute to provide an absolute standard.

Assuming kk is a field, the space of models would then be

(k\{0}×k×k)/(k\{0})k 2. (k \backslash \{0\} \times k \times k) / (k \backslash \{0\}) \cong k^2.

Would we be less inclined to drop the two massless particles model?

Posted by: David Corfield on June 17, 2009 10:48 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David wrote:

You’d think that having the unit of mass would be as though there were a third (massive) particle, kept in the vaults of a scientific institute to provide an absolute standard.

Exactly! In fact the standard kilogram is such a thing: an actual cylinder of platinum-iridium alloy, the International Prototype Kilogram, stored in a safe in a vault in the basement of a building owned by the International Bureau of Weights and Measures, in Sèvres, on the outskirts of Paris. I would go see it, but I wouldn’t be allowed to: three independently controlled keys are required to open the vault.

This is not a photograph of it, because nobody is allowed to take photos! It’s a computer-generated image. This raises an interesting question: at what point does protecting an entity mean that it might as well not even exist? But anyway…

Assuming kk is a field, the space of models would then be

(k\{0}×k×k)/(k\{0})k 2. (k \backslash \{0\} \times k \times k) / (k \backslash \{0\}) \cong k^2.

Exactly! There are other ways to reach this answer, but your way is quite nice.

Would we be less inclined to drop the two massless particles model?

Apparently so! Or more objectively speaking: it’s no longer weirdly different from the other models — it’s no longer a ‘stacky point’ in the moduli space, i.e. a model that has extra symmetries.

But now let me see if I can get someone to extract an important moral from your calculation. In my first puzzle we had a theory with one independent dimension: mass. The moduli space of models was something familiar from projective geometry. In my second puzzle we had a theory with two independent dimensions: mass and velocity. The moduli space of models was something familiar from biprojective geometry.

What about this time?

Posted by: John Baez on June 17, 2009 4:11 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

…stored in a safe in a vault in the basement of a building owned by the International Bureau of Weights and Measures, in Sèvres, on the outskirts of Paris…

That’s cool. I’m trying to think of a movie plot where the evil villain somehow breaks into that building and does something diabolical concerning the International Prototype Kilogram, maybe exchanges it for a fake one for some cunning reason I can’t think of right now. I can picture him saying while he does it “Fort Knox? That’s for tourists! The real deal is the International Bureau of Weights and Measures.”

Posted by: Bruce Bartlett on June 17, 2009 6:03 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

That’s part of the header of our universe in this multiverse. What is the topology of the multiverse? If you travel between the universes, how can you find your way home? At a minimum, all the fundamental constants must be the same, to at least the accuracy of International Bureau of Weights and Measures… Cf. Homer Simpson leaving the universe where it rains donuts…

Posted by: Jonathan Vos Post on June 17, 2009 7:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Bruce wrote:

I’m trying to think of a movie plot where the evil villain somehow breaks into that building and does something diabolical concerning the International Prototype Kilogram, maybe exchanges it for a fake one for some cunning reason I can’t think of right now.

I like this idea.

Don’t thieves steal famous paintings and sell them secretly to rich bastards who enjoy having them in their collections even if they can’t show these paintings to people without giving away their crime? I think I’ve even seen a movie where an art collector was also an art thief!

So surely we can imagine a crazy rich geek hiring someone to steal the International Prototype Kilogram.

I mean, Bill Gates bought the Codex Leicester, and Nathan Myrhvold, the Microsoft science advisor who earlier had a postdoc with Hawking, has built himself a copy of Charles Babbage’s Difference Engine:

So, there’s no telling what these computer nerds will do. One of them might want the International Prototype Kilogram in his collection! And then we’ll be stuck using one of the forty official replicas that were made in 1884.

Posted by: John Baez on June 18, 2009 2:25 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I’m trying to think of a movie plot where the evil villain somehow breaks into that building and does something diabolical concerning the International Prototype Kilogram, maybe exchanges it for a fake one for some cunning reason I can’t think of right now.

I wonder what the legal ramifications might be. Most (if not all) court systems that handle charges of commercial fraud, such as selling less than the weight stated on the package, ultimately defer (or claim to defer) to standards set up by states whose regulations refer to the International Bureau of Weights and Measures. Could you actually change the outcome if you could prove that International Prototype Kilogram was not the mass that people believed that it was? Or (more reasonably, but still pretty far fetched) could your lawyers bluff people into thinking that it would?

Posted by: Toby Bartels on June 18, 2009 10:49 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

John asks

What about this time?

Well, I guess k 2k^2 is pretty familiar in projective geometry as the space which just needs a line at infinity to make up the projective plane k 2k \mathbb{P}^2.

Posted by: David Corfield on June 17, 2009 9:39 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David said

… the space which just needs a line at infinity to make up the projective plane k 2k\mathbb{P}^2

The affine plane!

Sometimes called k𝔸 2k\mathbb{A}^2.

Posted by: Tim Silverman on June 17, 2009 10:35 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Great!

So, what’s the big moral lesson we learned from this puzzle?

Fill in the blank:

n=0n = 0: - - - - - - geometry

n=1n = 1: projective geometry

n=2n = 2: biprojective geometry

and say which parameter nn in our dimensional theory is controlling the sort of geometry relevant to its moduli space of models.

Posted by: John Baez on June 18, 2009 10:45 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Well, nn is the number of components in the grading object. I can’t remember what conditions you put on that object – an abelian group?

And n=0n = 0 geometry is affine.

Posted by: David Corfield on June 18, 2009 11:15 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Almost right! But ‘components in the grading object’ isn’t quite how a professional would say it.

Anybody: what is nn exactly? Say it like a mathematician! And why does the passage from n=0n = 0 to n=1n = 1 correspond to the historically important passage from affine to projective geometry?

Posted by: John Baez on June 18, 2009 12:15 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Does the grading object have to be a torsion-free finitely-generated abelian group? Then we would say nn is the rank.

Might it be nice to have some torsion? Bit odd I suppose – mass 29=1mass^{29} = 1.

Posted by: David Corfield on June 18, 2009 12:39 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

And why does the passage from n=0 to n=1 correspond to the historically important passage from affine to projective geometry?

i don’t completely agree with this way of describing the situation. arguably the historically important passage went in something closer to the opposite direction, and there may be better ways to analyze which kinds of dimensional theories correspond to which kinds of geometry.

as allen knutson pointed out here, projective varieties are roughly what you get from z-graded commutative algebras generated by elements in grade 1. let’s try to understand this in the context of the geometric interpretation of more general graded commutative algebras (aka “dimensional theories”).

like theories of other doctrines, a dimensional theory is generally built up in layers of stuff (in this case the generating dimensions), structure (the generating quantities), and properties (the generating equations).

the last stage in the process, the specification of the equations that the quantities are to obey, corresponds to the process of “carving out a subvariety of an ambient space” (or more generally, “carving out a substack of an ambient stack”). but will the ambient stack- the arena in which the geometry takes place- be affine space, projective space, or some other possibility? we wish to understand how that is determined by the first two stages of the process, the specification of the generating dimensions and of the generating quantities of the theory.

the basic idea is this: the dimensions of a dimensional theory are essentially the irreducible representations of its re-scaling group. when you specify the dimensions where the generating quantities of the theory are to live, you’re really specifying a representation of the re-scaling group, namely the direct sum of the corresponding irreducible representations. the “orbit stack” of this representation is the ambient stack in which the geometry lives.

(you can think of “orbit stack” here as a fancier version of “orbit space”. the orbit stack of a representation r of a group g is “the moduli stack for a g-torsor equipped with an element in r”. it’s sometimes important to treat g here as an “algebraic” group, functorially depending on the base ring over which you’re working.)

when your theory has just one generating dimension, that dimension itself corresponds to the “tautological” representation of the re-scaling group. projective n-space is (ignoring “the irrelevant point”) the orbit stack of the direct sum of n+1 copies of the tautological representation, so this explains allen’s comment about projective varieties arising from z-graded commutative algebras generated by elements in grade 1.

when a dimensional theory is “dimensionless”- in other words when the corresponding graded commutative algebra is “ungraded”, meaning graded by the trivial group- then its re-scaling group is also the trivial group. a representation of the trivial group is essentially just a vector space, and the orbit stack of that representation is, not too surprisingly, just the vector space itself, aka “affine space”. so of course john is right that affine varieties arise from the “ungraded” special case of graded commutative algebras.

i guess that the different emphasis that i’m trying to put on this is that whereas john is pointing out how the kind of geometry that you’re dealing with is partially determined by the rank of the grading group, i’m trying to explain how it’s more fully determined by the representation of the re-scaling group associated with the dimensions where the generating quantities live.

at the beginning of this post i threatened to argue that “the historically important passage went in something closer to the opposite direction”, but i haven’t gotten around to that yet; i’ll save it for another post.

Posted by: james dolan on June 20, 2009 11:11 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

As far as the history goes, it almost seems to have gone first one way and then back the other way! I was talking about the second phase, after Descartes’ ‘analytic geometry’, after Desargues had invented ‘points at infinity’, when projective geometry became a fashionable topic.

Posted by: John Baez on June 20, 2009 12:01 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Okay, let’s summarize where we are so far. We will start with some 2-category theory, but soon drift into high-school physics — so if you find this stuff scary, just keep reading!

A dimensional theory is a symmetric monoidal kk-linear category TT in which every object xx is a line object, meaning it has an ‘inverse’ x *x^*:

xx *I,x \otimes x^* \cong I ,

and it also has the special property that its self-braiding is trivial:

S x,x:xxxx S_{x,x} : x \otimes x \to x \otimes x

is the identity morphism.

Now, the pompously titled Fundamental Theorem of Dimensional Categories says that the 2-category of dimensional theories (described above) is equivalent to a 2-category in which an object is just a graded commutative kk-algebra AA with arbitrary abelian grading group GG.

Pretty soon I hope to write up the proof of this theorem. But it’s really easy to say how we get the abelian group GG from our dimensional theory TT. Here’s how: take the set of isomorphism classes of objects in TT. Tensor product of objects makes this set into an abelian group GG.

So now I can answer David’s question:

Does the grading object have to be a torsion-free finitely-generated abelian group?

No. But you’re on the right track, because in the most familiar cases it is!

A slightly less pedantic name for this kind of group is a free abelian group on nn generators.

A still less pedantic name for it is n\mathbb{Z}^n.

A dimensional theory with grading group n\mathbb{Z}^n is the same as a theory of physics in which there are:

  • nn independent dimensions D 1,,D nD_1, \dots, D_n, all others being products of powers of these, and
  • a set of basic quantities each having a specified dimension D 1 d 1,,D n d nD_1^{d_1} , \dots, D_n^{d_n}, satisfying
  • homogeneous polynomial equations.

For some reason — why? — many physicists like n=3n = 3, with

D 1=lengthD_1 = length D 2=timeD_2 = time D 3=massD_3 = mass

They often consider theories with fewer than 3 dimensions, built by the trick I described in my last puzzle. A bit more rarely, they use theories with more than 3 units: temperature and electric charge are popular candidates for a fourth one.

Fun puzzle: guess how many dimensions the official International System of Units — the so-called ‘SI system’ — has!

Might it be nice to have some torsion? Bit odd I suppose — mass 29=1mass^29=1.

Excellent suggestion!

While it’s definitely a bit exotic, I think it would be very fun and possibly educational to consider a dimensional theory where the grading group had some torsion.

Suppose for example that we take Prime Minister Berlusconi as our fundamental unit of volume. Then we can set length 3=1length^3 = 1 if we like. What kind of moduli spaces for theories do we get then?

Posted by: John Baez on June 18, 2009 2:01 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

When you say

…arbitrary abelian grading group GG,

it sounds like you’re allowing non-finitely generated groups. So we can have G=,/G = \mathbb{Q}, \mathbb{Q}/\mathbb{Z} or /\mathbb{R}/\mathbb{Q}, etc.?

Posted by: David Corfield on June 18, 2009 2:38 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

When you say

…arbitrary abelian grading group G,

it sounds like you’re allowing non-finitely generated groups. So we can have G=ℚ,ℚ/ℤ or ℝ/ℚ, etc.?

we’ve already been talking about this, over here and over here for example.

by the way, this business of how the “divisors” on a variety or scheme form the dimensions of a dimensional theory, particularly in the case of a riemann surface or of the spectrum of a ring of algebraic integers, is fairly easy to understand (much moreso than you might think from the traditionally obscurantist ways in which these ideas are often presented), and crucial to understanding what weil was really talking about in his letter to his sister.

“divisor” is really just another item in our list of concepts (“dimension”, “line”, “line object”, “line bundle”, “1-dimensional vector space”, “invertible vector space”, “invertible sheaf”, and so forth) that are all secretly more or less the same thing.

also by the way, you should keep in mind that since a dimensional theory is a category, there’s a bit of ambiguity in referring to its “group of objects” (or “grading group”). do you mean the group of objects up to strict equality, or up to isomorphism? the former is the actual grading group, the latter is the grading group modded out by the subgroup consisting of the “trivialized dimensions” (those for which a standard unit is kept in the vaults of the intercosmic bureau of standards and units). i discussed this a bit over here.

whichever one you meant, the group of dimensions or the group of “independent dimensions”, both of them can get pretty big in useful examples.

Posted by: james dolan on June 19, 2009 1:45 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Jim Dolan wrote:

“…the traditionally obscurantist ways…”

Sorry for butting in again without following the discussion. But when I noticed this phrase, I couldn’t resist registering an objection. To put matters this way is good fun sometimes and can give a person the psychological boost necessary to attempt better expositions. But my impression is it’s overdone.

According to the OED:

——————————————
obscurantism

/obskyoorantiz’m/

• noun the practice of preventing the facts or full details of something from becoming known.
——————————————

I know plenty of algebraic geometers who are obscure. But frankly, I don’t anyone who is deliberately so.

Posted by: Minhyong Kim on June 19, 2009 5:42 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

also by the way, you should keep in mind that since a dimensional theory is a category, there’s a bit of ambiguity in referring to its “group of objects” (or “grading group”). do you mean the group of objects up to strict equality, or up to isomorphism? the former is the actual grading group, the latter is the grading group modded out by the subgroup consisting of the “trivialized dimensions” (those for which a standard unit is kept in the vaults of the intercosmic bureau of standards and units). i discussed this a bit over here.

whichever one you meant, the group of dimensions or the group of “independent dimensions”, both of them can get pretty big in useful examples.

the group of nominal dimensions modulo the subgroup of trivialized ones should probably be called something like the “effective” dimensions rather than the “independent” ones, since “independent” suggests that they form a basis. so when there are n independent dimensions, the group of effective dimensions is z^n.

Posted by: james dolan on June 20, 2009 6:20 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Since a dimensional category is a very general concept, the corresponding kind of algebra is graded by a very general sort of abelian group: any abelian group.

But algebraic geometry traditionally focuses on reasonably ‘small’ algebras — for example, finitely generated ones. For something like this, an infinitely generated grading group is overkill. So let’s not worry about those huge abelian groups yet.

On the other hand, finite grading groups might be worth pondering. So: did anyone think about how having a unit of length 3{}^3 is different from having a unit of length?

(If it helps, remember our ‘lengths’ are taking values in an arbitrary field kk.)

Posted by: John Baez on June 18, 2009 7:04 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

So we don’t necessarily have cube roots.

Let’s have a theory with two particles, each with a length and a unit of volume.

So

l 1:1length l 2:1length \begin{aligned} l_1 : 1 \to length \\ l_2 : 1 \to length \end{aligned}

and a Berlusconi

b:1length 3, b : 1 \to length^3,

which is invertible.

Without the Berlusconi we would have had moduli space k 1k \mathbb{P}^1. Had it been a unit of length 1length^1, we would have had k 2k^2.

Let’s have kk be the field with 7 elements, and bb can be 2. If we consider two models

l 1=0,l 2=3 l 1=0,l 2=6, \begin{aligned} l_1 = 0, l_2 = 3 \\ l_1 = 0, l_2 = 6, \end{aligned}

they would have been identified without a unit, and would have been counted as different with a unit of length.

But in the case as described we shouldn’t distinguish them since a natural transformation between models which doubles lengths will leave volumes alone.

On the other hand, writing models as (l 1,l 2,b)(l_1, l_2, b), (0,6,2)(0, 6, 2) and (0,3,1)(0, 3, 1) won’t be the same model.

Anyway there’s an equivalence relation on k×k×k\{0}k \times k \times k \backslash \{0\} with classes

{(αm,αn,α 3p|αk\{0}}. \{(\alpha \cdot m, \alpha \cdot n, \alpha^3 \cdot p | \alpha \in k \backslash \{0\}\}.

Posted by: David Corfield on June 19, 2009 10:01 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David wrote:

… it sounds like you’re allowing non-finitely generated groups. So we can have G= ,/,/\mathbb{Q}, \mathbb{Q}/\mathbb{Z}, \mathbb{R}/\mathbb{Q}, etc.?

John wrote:

… algebraic geometry traditionally focuses on reasonably ‘small’ algebras — for example, finitely generated ones. For something like this, an infinitely generated grading group is overkill.

Whoops! I was being stupid, seduced by my physics puzzles and the tendency of algebraic geometers to talk a lot about \mathbb{Z}-graded algebras. As Jim pointed out, there are plenty of dimensional theories in algebraic geometry where my ‘group of dimensions’ — the group of isomorphism classes of objects — is very large. And we’ll have to get into those soon.

For example, take a variety XX and let our dimensional theory be the category of line bundles over XX. Then its group of dimensions is the group of isomorphism classes of line bundles over XX. In the 90’s, fans of the Star Trek: the Next Generation conducted a letter-writing campaign to persuade mathematicians to name this the Picard group, in honor of an episode where the good captain was trying to prove Fermat’s Last Theorem with the help of this concept. The name seems to have stuck.

When XX is a complex curve, the connected component of the Picard group is called the the Jacobian variety. It’s usually uncountably generated as an abelian group. For a complex curve of genus gg, it’s the product of 2g2g copies of the circle. The Picard group is this times \mathbb{Z}.

Note, the Picard group is the group of isomorphism classes of algebraic, or if you prefer, holomorphic line bundles over XX. The group of isomorphism classes of topological line bundles over a complex curve is just \mathbb{Z}; the Jacobian variety consists of the line bundles that are topologically isomorphic, but not holomorphically.

So, if our dimensional theory was the category of topological line bundles over a complex curve, the biggest grading group we could sensibly use is \mathbb{Z}. That group is still used a lot in the holomorphic case, but we can also use a much larger group.

Posted by: John Baez on June 20, 2009 10:25 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David wrote approximately:

Let’s have a theory with two particles, each with a length, and a unit of volume.

So

l 1:1length l 2:1length \begin{aligned} l_1 : 1 \to length \\ l_2 : 1 \to length \end{aligned}

and a Berlusconi

b:1length 3, b : 1 \to length^3,

which is invertible.

Without the Berlusconi we would have had moduli space k 1k \mathbb{P}^1. Had it been a unit of length 1length^1, we would have had k 2k^2.

[…]

Now there’s an equivalence relation on k×k×k\{0}k \times k \times k \backslash \{0\} with classes

{(αx,αy,α 3x|αk\{0}}. \{(\alpha \cdot x, \alpha \cdot y, \alpha^3 \cdot x | \alpha \in k \backslash \{0\}\}.

Great! The Berlusconi variety takes its rightful place alongside the Albanese variety, the Veronese embedding and the Segre embedding!

I’m not sure in what sense it’s a ‘variety’. But anyway, it’s a moduli space M 3M_3 that’s in some sense intermediate between the projective line and the affine plane. And we can build such a moduli space M dM_d assuming we have a unit of dimension length dlength^d for any kk, not just d=3d = 3.

It looks like this: take k 3{0}k^3 - \{0\}, where kk is our field, and mod out by the equivalence relation

(x,y,z)(αx,αy,α dz)(x,y,z) \sim (\alpha x, \alpha y, \alpha^d z) where α0\alpha \ne 0.

This space is probably already familiar to lots of people. Anyone here know it?

Posted by: John Baez on June 20, 2009 10:38 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Fun puzzle: guess how many dimensions the official International System of Units — the so-called ‘SI system’ — has!

I wouldn't ask it that way. The principle behind the SI system is that nature itself has 77 dimensions; the purpose of the SI system is then to reduce these 77 to 00 by assigning 77 units.

Actually, the dimension of luminous intensity (whose SI unit is the candela) is not really a purely physical concept at all, but depends on the physiology of the human eye; so let's ignore that one. (Actually, they use a standardised model of the human eye, in theory irrespective of actual humans, so we could consider it purely physical. But let's not make things too complicated.)

So the 66 dimensions are [l][l] (length), [t][t] (time interval), [m][m] (mass), [I][I] (electrical current), [T][T] (temperature), and [n][n] (amount of substance); their SI units are m\m (metre), s\s (second), kg\kg (kilogramme), A\A (ampere), K\K (kelvin), and mol\mol (mole).

But the mole is such a silly unit, isn't it? Sure, in practice we use it since we don't know exactly what Avogardo's number is, but in theory it's just a number, like a dozen. (It's presumably not precisely an integer, but that doesn't matter.) So instead of using the mole to reduce the number of dimensions, let's use the ‘natural’ unit of amount of substance that consists of 11 item.

And if you understand special relativity, then you know that lengths and time intervals are different aspects of the same fundamental concept (spacetime interval). Even the BIPM recognises this and defines the metre so that the second is (in ‘natural’ units) exactly cs=2.99792458×10 8mc \s = 2.997\,924\,58 \times 10^8 \m. So the metre is not so fundamental after all!

And if you understand quantum mechanics, then you know that energy (equivalent to mass by the previous paragraph) is really the same as frequency, the inverse of time interval. When the BIPM gets sick of supervillains messing with the IPK, then they might adopt a proposal to (effectively) define Planck's constant to be exactly 6.62606896×10 34kgm 2/s{6.626\,068\,96 \times 10^{-34} \kg \m^2}/{\s}. Then both the metre and the kilogramme will be defined in terms of fundamental physics and the second.

On the other hand, if you understand universal gravitation (Newton's theory is enough, although Einstein's can reassure you that this is very fundamental), then you know another fundamental universal constant of physics whose SI unit involves only the second, the metre, and the kilogramme (whence now, only the second). Although we can't measure it precisely enough to make a practical unit of measurement out of it, in theory we now have a ‘natural’ unit of time interval. So the 33 usual dimensions are gone, without any obviously arbitrary choices!

We've already done amount of substance, so let's quickly polish off the last 22 allegedly fundamental dimensions: [I][I] is handled by the remaining constant in Maxwell's equations (or even Coulomb's law of electrostatics), and [T][T] is handled by the ideal gas constant (or more fundamentally, by Boltzmann's constant relating entropy to statistics).

Suppose for example that we take Prime Minister Berlusconi as our fundamental unit of volume. Then we can set length 3=1length^3 = 1 if we like.

Ah, but the physicist in me knows that a unit of volume gives me a unit of length: the unique cube root of the given volume. Since, as I mentioned before, most of our dimensions are actually oriented lines, we can also take principal square roots if we need them. Indeed, much of my discussion just above requires taking square roots when you work out the details.

In fact, I see that we must take square roots not only to get the natural units of [l][l], [t][t], and [m][m] (which is OK since those dimensions are oriented) but again to get the natural unit of electric current. That dimension is not oriented, so we do have to make an arbitrary choice: whether the charge on the electron is positive or negative.

This all means that we're really working with n\mathbb{Q}^n and not n\mathbb{Z}^n after all. Indeed, I have seen people analyse dimensional analysis using n\mathbb{Q}^n, basically for this reason. In fact, I've always wanted to use n\mathbb{R}^n as the group of dimensions, but I don't think that there's actually any physical reason to do so. (I know that you want to use n\mathbb{Z}^n for algebraic geometry; I'm just saying that n\mathbb{Q}^n is what really shows up in physics.)

Posted by: Toby Bartels on June 19, 2009 12:52 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Toby wrote:

When the BIPM gets sick of supervillains messing with the IPK, then they might adopt a proposal to (effectively) define Planck's constant to be exactly 6.62606896×10 34kgm 2/s{6.626\,068\,96 \times 10^{-34} \kg \m^2}/{\s}. Then both the metre and the kilogramme will be defined in terms of fundamental physics and the second.

Another fun proposal, discussed by Stefan over at the blog Backreaction, is to define the kilogram to be the mass of a specific number of atoms of silicon.

(You’ll notice that Stefan outdid the authors of Wikipedia: for this blog entry, he snuck into the vault and photographed the International Prototype Kilogram!)

In fact, I see that we must take square roots not only to get the natural units of [l], [t], and [m] (which is OK since those dimensions are oriented) but again to get the natural unit of electric current. That dimension is not oriented, so we do have to make an arbitrary choice: whether the charge on the electron is positive or negative.

And of course we make the convenient choice: it’s negative.

This all means that we’re really working with n\mathbb{Q}^n and not n\mathbb{Z}^n after all. Indeed, I have seen people analyse dimensional analysis using n\mathbb{Q}^n, basically for this reason.

Okay, that’s interesting. It leads us away from ‘traditional’ algebraic geometry towards a kind of algebraic geometry where we are free to take the nnth root of any quantity we like… but this subject may still fit comfortably under the big umbrella of ‘dimensional theories’.

I don’t know much about this subject — surely it’s been studied! All that leaps to my mind is Puiseux series.

Posted by: John Baez on June 20, 2009 11:40 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Another fun proposal, discussed by Stefan over at the blog Backreaction, is to define the kilogram to be the mass of a specific number of atoms of silicon.

Yes, I find that proposal more interesting, really, but it didn't fit into my story as well.

And of course we make the convenient choice: it’s negative.

That page has a nice quotation:

“The nice thing about standards is that there are so many of them to choose from.”

And the bottom of the page reminds me that the bad thing about copyleft licences is that there are so many of them to choose from.

Posted by: Toby Bartels on June 20, 2009 9:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

This all means that we’re really working with ℚ^n and not ℤ^n after all. Indeed, I have seen people analyse dimensional analysis using ℚ^n, basically for this reason.

Okay, that’s interesting. It leads us away from ‘traditional’ algebraic geometry towards a kind of algebraic geometry where we are free to take the nth root of any quantity we like… but this subject may still fit comfortably under the big umbrella of ‘dimensional theories’.

i’m not sure that i understand exactly what toby’s concerns are here, but i suspect that john might be right about not having to go outside the world of dimensional theories; after all, q-modules are special cases of z-modules.

Posted by: james dolan on June 22, 2009 9:35 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

After some intense instruction in ‘algebraic geometry for category theorists’ by John in a Parisian café on Wednesday, I was visited by a thought or two the next day.

First, something that really helped was an explanation of why it is a good thing to consider all line bundles over a space, rather than functions (or sections of the trivial line bundle) on that space. The example went via the problem that the only holomorphic functions on the Riemann sphere are the constant ones.

Then a couple of speculations. If we can think of the rationals \mathbb{Q} as sections of a line bundle (in \mathbb{Z}) over the primes (or spec()spec(\mathbb{Z})), are there other such line bundles? Is there a story in terms of graded commutative rings?

Given that dimensional categories are equivalent to GG-graded commutative rings for some abelian group GG, and given that these have been related to classical physical theories, are we going to see a generalisation to the noncommutative case? I see that at least one form of noncommutative line bundle exists.

Hmm, if you go around subtracting products of position and momentum operators as in the commutator relations, doesn’t it suggest the grades of these quantities are abelian?

Aha, we have an entry on noncommutative algebraic geometry. Then there’s the worry of which noncommutative geometry?

Posted by: David Corfield on July 17, 2009 10:03 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

In the case of number fields (that is, the case of spectra of rings of algebraic numbers) you get most easily to (sections of) line bundles via divisors, which in this case means, basically, fractional ideals. Sections of the trivial bundle, corresponding to principal divisors, then means, in this case, principal ideals.

The algebraic integer rings themselves of course correspond to the affine case; to get something complete, analogous to projective spaces, you need to add on the analogue of points at infinity, viz. the real prime (in the case of \mathbb{Z}) or assorted real and complex primes at infinity (in the more general case).

I’ve been meaning to write something about this anyway, so perhaps this is an opportunity for me to get something written up in a less cryptic way.

Posted by: Tim Silverman on July 17, 2009 1:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I’ve thought about this myself, but I’m not too convinced. In ordinary algebraic geometry, say where we complete affine space to projective space, there is a certain homogeneity where every point in projective space looks like every other point, whereas in the completion you’re proposing, the archimedean places “at infinity” are qualitatively different from the nonarchimedean ones (the latter are ultrametric, for example).

I mean, maybe you’re right and the analogy is a really good one, but I don’t see that as yet. Could you elaborate?

Posted by: Todd Trimble on July 17, 2009 4:03 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Could you elaborate?

Not necessarily very intelligently—I’m trying to repeat conventional wisdom more than I’m trying to justify it, and quite possibly I’m repeating it wrong.

However, in defense of the analogy, I would point out that, to my mind, all places in number fields tend to have their own distinctive character in a way that is quite unlike the behaviour of points in algebraic varieties. (For instance, the sets of values that algebraic integers can take at different places are different from each other, quite unlike functions on an affine or projective space.) So the places at infinity aren’t necessarily more different than the finite places.

But I am not in any way an expert on this, so I don’t think I have anything deep to say, I’m afraid. There are others out there far more qualified to comment.

Posted by: Tim Silverman on July 17, 2009 4:38 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Tim wrote:

all places in number fields tend to have their own distinctive character in a way that is quite unlike the behaviour of points in algebraic varieties.

I think a lot could be said there, but I guess what I’d be interested in is: in what sense is “at infinity” for archimedean places like “at infinity” for projective completions? Mind you: I’d like a convincing argument for that! It’s just that I don’t have one myself.

Posted by: Todd Trimble on July 17, 2009 6:36 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

They’re archimedean! What more could you want?

OK, if I’ve understood what you’re asking for, the argument is this:

On \mathbb{Q}, the non-archimedean valuations look like n-n for a rational p nap^n a, prime pap\nmid a. On k(X)k(X) for some field kk, the non-archimedean valuations look like n-n for a rational function P nQP^n Q, irreducible PQP\nmid Q, and that corresponds to the point where PP has its zero.

On k(X)k(X), the archimedean valuation of a rational function P(X)Q(X)\frac{P(X)}{Q(X)} is deg(P)deg(Q)deg(P)-deg(Q).

Now, suppose we have P(X)=a 0+a 1X++a mX mP(X)=a_0+a_1 X+\cdots+a_m X^m and Q(X)=b 0+b 1X++b nX nQ(X)=b_0 + b_1 X +\cdots+b_n X^n, so deg(P)=mdeg(P)=m and deg(Q)=ndeg(Q)=n. Now set Y=1/XY=1/X.

Then P(X)Q(X)=Y nma 0Y m+a 1Y m1++a mb 0Y n+b 1Y n1++b n\frac{P(X)}{Q(X)}=Y^{n-m}\frac{a_0 Y^m+a_1 Y^{m-1}+\cdots+a_m}{b_0 Y^n+b_1 Y^{n-1}+\cdots+b_n}, so, taking YY as an irreducible polynomial, the valuation at YY would be mn=deg(P)deg(Q)m-n=deg(P)-deg(Q). But YY is the polynomial which is 00 at \infty. So this is the valuation at \infty. So in this case the “at infinity” for the archimedean place is exactly the “at infinity” for the projective completion. Better still, the absolute valuations at infinity for both R=R=\mathbb{Z} and R=𝔽 q[X]R=\mathbb{F}_q[ X] are |x| =card(R/(x))\vert x\vert_\infty=card(R/(x)). So the case of \mathbb{Z} (or, hence, algebraic integer rings generally) is directly analogous to the function field case; the fact that the product of absolute values is 11 is the “same” fact as that all principal divisors are zero divisors, etc.

(I’ve lifted this argument a bit sloppily from Yves Hellegouarche’s Invitation to the Mathematics of Fermat-Wiles. If it’s already familiar to you, and you were asking for something more, then I apologise. Hopefully it will be useful to somebody, anyway.)

Posted by: Tim Silverman on July 17, 2009 7:49 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Yeah. I remember taking almost exactly the same line once when I was a graduate student, and I remember being firmly silenced by my teacher Jim Cogdell in response. I’ll try to recall the substance behind the silencing.

I think the trouble is that there is no absolute “ring of integers” sitting inside the function field k(x)k(x) with which you could point to some place as being “infinite”, for the simple reason that the field k(x)k(x) has lots of automorphisms (fractional linear transformations). It’s true that algebraic number fields can also have automorphisms, but these automorphisms take archimedean places to archimedean places and nonarchimedean places to nonarchimedean places, and never the twain shall meet. So for function fields with a high degree of homogeneity, like the function field for the projective line, it’s somewhat illusory to say that some place is “infinite”, unless you fix at the outset some integral subring k[x]k[x] inside. Whereas “infinite” in the sense of “archimedean” has a more absolute character.

So, if I’m not mistaken: projective completion for an affine curve allows you to pass from the coordinate ring of the curve to the function field, and then look at discrete valuation rings inside that as points of the projective completion, but once you pass to the function field and forget the coordinate ring it came from, there is no particular sense of what’s “infinite” and what’s not. Now I guess it’s true that we sort of do the same thing with rings of algebraic number fields: we can pass to the field of fractions and ask what are its valuations (up to equivalence). In that sense I’d agree with you. But in another sense we’re doing something different: it’s not just DVR’s in the field we’re considering as in the function field case – there’s something genuinely “extra” about the archimedean places that make the completion process you’re proposing for algebraic number fields a crucially important but nevertheless different kind of completion than for function fields, and the sense of “infinite” receives different types of explanations for these two cases.

Did I make sense here? My guess is that there is actually agreement between us, masquerading as disagreement (to echo something another Tim – Tim Gowers – wrote recently).

Posted by: Todd Trimble on July 17, 2009 9:56 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Yes, this makes sense to me. I think I might phrase the matter slightly differently, I guess. That is, for function fields, given a coordinate ring (for an affine scheme), you get a definition of what it means to throw in points at infinity to give the projective closure; and it means the same thing in the case of number fields, except that you don’t get to choose the ring, it’s put on the plate in front of you, and you have to use it whether you like it or not. So, the way I’d think of things, “at infinity” is relative in one case, and absolute in the other, but that’s not due (in my mind) to different concepts of “at infinity”, but due to the same concept being applied in different contexts. But of course this is a matter of interpretation rather than substance.

I’m chuffed to see that my first reply (the bit you quoted) still seems to be pertinent: yes, in \mathbb{Z}, the point at infinity is special and uniquely identifiable—but the property of being special and uniquely identifiable is not, itself, a special property of the point at infinity, because all the other points are special (in their own way) too! That’s my current take on things, anyway.

Posted by: Tim Silverman on July 17, 2009 10:33 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I think the trouble is that there is no absolute “ring of integers” sitting inside the function field k(x) with which you could point to some place as being “infinite”, for the simple reason that the field k(x) has lots of automorphisms (fractional linear transformations). It’s true that algebraic number fields can also have automorphisms, but these automorphisms take archimedean places to archimedean places and nonarchimedean places to nonarchimedean places, and never the twain shall meet. So for function fields with a high degree of homogeneity, like the function field for the projective line, it’s somewhat illusory to say that some place is “infinite”, unless you fix at the outset some integral subring k[x] inside. Whereas “infinite” in the sense of “archimedean” has a more absolute character.

just a minor comment here: it seems possible to question whether it’s the highly non-homogeneous spaces or perhaps instead the highly homogeneous ones that are the odd men out here. lots of ordinary algebraic varieties have special points such as weierstrass points that stick out like a sore thumb. i don’t remember offhand thinking much about how the specialness of such points might compare to the specialness of archimedean places of number fields.

Posted by: james dolan on July 17, 2009 11:10 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Yes, I was thinking about that as well, and I don’t mean to insist too much on “homogeneity” as drawing the line I was attempting to draw. It was kind of a cheap way of making a point that unfortunately I don’t understand too well: even in the cases where all primes have their own special individual flavor, there’s just something “extra-special” about archimedean places, that sets apart the whole study of global number fields from the study of curves over 𝔽 q\mathbb{F}_q (or over an algebraic closure of that), and it doesn’t seem too crazy to insist on the difference. You know, the difficulty of the Riemann hypothesis over global number fields, that sort of thing.

Posted by: Todd Trimble on July 18, 2009 12:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Now I think it is my turn to ask you to elaborate, as I am not really convinced yet. I am certainly very open to persuasion and would not place any great weight on my intuitions here, for fear they would splinter underfoot, but my instinct generally is to avoid marking out one case as particularly special unless strong evidence supports it. I can see (of course!) that number fields are in important respects very different from function fields, and I find the idea that the archimedean places play a special role in this an interesting one, but as yet I do not see it. I’d be glad of any more insight you can give me into this. My own ideas are really quite unformed, and my intuition needs training.

Posted by: Tim Silverman on July 18, 2009 9:47 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Hi Tim. I somewhat doubt I can be very persuasive, but one thing that comes to mind that really sets archimedean places apart (to my mind) is the radically different natures of the local completions. For nonarchimedean places, the topology of the local fields is totally disconnected: they are (filtered colimits of) Cantor spaces, or more precisely filtered colimits of profinite completions. This would have all sorts of consequences; for example, Fourier analysis, Bruhat-Schwartz spaces, and so on are treated very differently in the archimedean as opposed to nonarchimedean cases (more below).

I think this is connected with another basic schism: a place is nonarchimedean if and only if it is discrete or ultrametric.

Another schism that comes to mind is the appearance of Euler factors for zeta functions. I don’t understand this well, but my rough understanding is that the natural setting for things like the functional equation from the point of view of Tate’s thesis is adelic. Taking the case \mathbb{Q}, the factor (1p s) 1(1 - p^{-s})^{-1} in the Euler product

ζ(s)= p(1p s) 1\zeta(s) = \prod_p (1 - p^{-s})^{-1}

then pops out from the study of Fourier analysis on the local completion p\mathbb{Q}_p. To complete the picture, one should also incorporate a corresponding Euler factor at the archimedean place, which has the form

E (s)=π s/2Γ(s/2)E_{\infty}(s) = \pi^{-s/2}\Gamma(s/2)

when one carries out the analogous Fourier analysis. (Then the full product

Z(s)= placesvE v(s)Z(s) = \prod_{places v} E_v(s)

satisfies the functional equation Z(s)=Z(1s)Z(s) = Z(1-s), as a byproduct of some Fourier duality statement on the locally compact ring of adeles in which the global field is discrete and cocompact, like Poisson summation.) My thinking is that while the Euler factor at infinity clearly plays an important role in all this, its basic analytic character just looks to me markedly different from that of the other Euler factors. I’ll remark in this connection that its presence is connected with the presence of the “trivial zeroes” of the zeta function, whereas the other Euler factors aren’t.

(I think the very different appearance of E (s)E_{\infty}(s) might be again traceable to the nondiscreteness of the valuation. In the nonarchimedean case, the Euler factor E p(s)E_p(s) expands as a geometric series, where each summand p nsp^{-n s} corresponds to a value p np^{-n} of the discrete valuation.)

I guess I’ll present all this as prima facie evidence of my intuition that archimedean places are “extra-special” or in a class of their own (freely admitting that my own intuition doesn’t count for much here – this isn’t my area of mathematics). I think it’s at least true that what happens at archimedean places receives its own separate analysis in many cases, and if one scans some of the literature, one often sees hypotheses like “let SS be a finite set of places which includes all the infinite places” – further evidence that finite places are treated separately apart from infinite places.

Finally, I might have made it look as if I were laying the greater difficulties in the number field case as opposed to the function field case (as in the case of RH) at the feet of the presence of archimedean places. I don’t have much firm support for such a thesis. I guess it’s well known that there are geometric interpretations in the function field case (permitting cohomological techniques) which aren’t really there in the number field case, but I couldn’t say that’s all the fault of infinite places! I did find a brief passage on page 45 (46 of 90) of this paper by Connes which illustrates one sort of difficulty with archimedean completions, but I would count that evidence as relatively slim in the scheme of things.

Maybe someone with greater expertise could weigh in?

Posted by: Todd Trimble on July 18, 2009 2:02 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Thanks for this, Todd. I understand what you’re saying. Obviously I was not completely oblivious to all these points; I guess my instinctive feeling—for what little it’s worth—was always that, if only I could look at these things in the right way, these differences wouldn’t seem like such a big deal. But this is backed up by no better evidence that that infinite places are obviously not completely different from finite places (including similarities which arise in unexpected ways) so there must be more similarity than meets the eye. Which is pretty weak!

Anyway, this has reminded me that I really need to try to understand this stuff better, so thanks for that.

Posted by: Tim Silverman on July 18, 2009 6:49 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Tim – you could very well be right, and I’d be happy to discuss it again some time. It would be absurd for me to hold hard and fast opinions here!

Posted by: Todd Trimble on July 18, 2009 10:19 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

As Tim has been saying, if O is the ring of integers in a number field, it is a common view-point in number theory to regard the addition of the archimedean places to Spec O as a kind of completion of O, analogous to adding points at infinity in projective geometry.

I will explain one basic motivation for this in a moment, but first, let me explain how to canonically normalize the absolute values for each of the places: if F is the field of fractions of O (so, a number field) and v is a place of F, then F_v (the completion of F at v) is a locally compact field, and so has an additive Haar measure. If a is non-zero element in F, than multiplication by a scales Haar measure by some amount; this is defined to be |a|_v (the v-adic absolute value of a). If v is a real place, so that F_v = R (the real numbers), then this is the usual absolute value. If v is a complex place, so that F_v = C (the complex numbers) then this is the square of the usual complex absolute value. If v is a p-adic place, so that F_v is a discretely valued field with residue field of order q, and a has order n with respect to the discrete valuation on F_v, then |a|_v = q^{-n}.

Now, with this normalization understood, we have the following theorem: if a is a non-zero element of F, then

product over all v of |a|_v = 1 .

This makes equally good sense if F is the
function field of a curve in characteristic p, and in this case, taking logs of both sides of this equation one obtains the usual statement that the sum of the orders of all zeroes and poles of a non-zero rational function on a curve is equal to zero. This latter statement is a well-known property of projective smooth curves which crucially relies on the curve being projective. Similarly, the product formula in the number field case will be false if we don’t include the archimedean places, so this formula gives some (philosophical, if you like) evidence that adding the archimdedean points is like completing Spec O.

The analogy goes much further: suppose that X is a smooth projective curve over the number field F, and extend X (as one can) to a regular proper scheme X’ flat over Spec O. (So one can think of X’ as being like a smooth “surface” mapping properly down to the “curve” Spec O.) If x and y are two points of X over F, then we may take their Zariski closures to get divisors x’ and y’ lying in X’. By analogy with the usual theory of surfaces, we could then try to develop an intersection theory for divisors in X’, and in particular, apply it to x’ and y’. (The value of this intersection pairing on x’ and y’ would describe how congruent x and y become modulo various primes of O.)

Now, usual intersection theory only works well on projective surfaces (in the sense that if we want the intersection number to depend only on the rational equivalence class of the two curves being intersected, say, then the surface had better be projective). It turns out that to get a good intersection theory on X’, we similarly have to complete it, by adding fibres X’_v over the various archimedean places v of F.

Once we do this, we get a good intersection theory. It is called “arithmetic intersection theory”, and more generally, the theory that results by pursuing the idea of adding archimedean places to Spec O is called Arakelov theory. It plays an important role in various parts of number theory; one of the first major applications was Faltings use of it in his Fields medal-winning proof of the Mordell conjecture. Some of the results in the theory include an “arithmetic Riemann-Roch theorem”, which relates arithmetic intersection theory to the “arithmetic Chern classes” of “arithmetic line bundles”.

One interesting feature of Arakelov theory is that the geometry over the archimedean places often involves quite serious harmonic analysis. For example, in the intersection theory for x’ and y’, when you think of them as points in X(F_v) (which is the fibre of X’ over an archimedean place v), they are just a pair of points x and y in a Riemann surface, and so don’t intersect in any obvious way. Never the less, these two points do make a contribution to the intersection pairing; one computes it using Green’s functions and related ideas. (To be precise, if you want an answer independent of choices, you shouldn’t work with individual points x and y, but rather, with linear combinations of points, i.e. divisors, of degree zero.)

Incidentally, regarding a comment of Todd’s above, it is often the case that people choose “ a finite set S of places which includes all the infinite places”. But this doesn’t necessarily mean that much; often, the set S is taken to be large enough as necessary for whatever purposes are at hand, but finite; since there are only finitely many archimedean places, it is then usually no trouble to include them in S, and so people often do.
(Typically, S is a set of places where ramification is allowed to occur, and ramification at an archimdean place is not normally as serious as at a finite place; it just means that a real place extends to a complex place.)

For some delicate questions, the precise members of S matter, and then the archimdean places are not always included. For example, the Hilbert class field of a number field F is the maximal abelian extension unramified at *every* place, including the archimedean ones. (Its Galois group over F is canonically isomorphic to the class group of O.) For example, Q[\sqrt{3}] has class number 1, and so has trivial Hilbert class field. On the other hand, Q[\sqrt{-1},\sqrt{3}] is a degree two extension of Q[\sqrt{3}] which is unramified at every finite place (It is only ramified at the real place.) More generally, in class field theory, the archimedean places come into play on an equal footing to the non-archimedean ones.

On the other hand, Todd is correct that the arguments and structures at the archimedean places are often quite different to what happens at the non-archimedean places. (Gamma factors look different to usual Euler factors; Green’s functions are quite different to congruences.) Still, one can often find a framework in which the structures at all the places can be described in some reasonably uniform manner. (As a basic example, despite the fact that archimedean and non-archimedean absolute values are quite different, they both fit into the same general notion of absolute value, and their canonical normalizations have a common description, as given above, in terms of how they affect Haar measure.)

Posted by: Matthew Emerton on July 19, 2009 3:50 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

If v is a real place, […] then this is the usual absolute value. If v is a complex place, […] then this is the square of the usual complex absolute value.

That's wild! I've known people to use the usual absolute value, and I've known people to prefer its square, but I've never before known people to mix them like this. But I see how it happens here, coming from a scaling of Haar measure on the appropriate field.

(Sorry for the perhaps rather trivial comment, but this is the first time in a while that I've really understood something in this thread!)

Posted by: Toby Bartels on July 19, 2009 6:23 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Q/Z is part of the theory of Brauer groups (please see the wikipedia entry on Brauer group), and Brauer groups are intimately related to gerbes.

Also, Michael Murray’s concepts about bundle gerbes are derived from starting with line bundles.

It has even been shown that a gerbe defined over a number field is related to the the Brauer-Manin construction (but I am not now able to find via google the full paper about this).

Posted by: Charlie Stromeyer on July 17, 2009 4:25 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

It was a lot of fun explaining my project with Jim Dolan to David, since I haven’t really tried explaining the whole thing to anyone before.

Some interesting questions:

First, something that really helped was an explanation of why it is a good thing to consider all line bundles over a space, rather than functions (or sections of the trivial line bundle) on that space. The example went via the problem that the only holomorphic functions on the Riemann sphere are the constant ones.

Then a couple of speculations. If we can think of the rationals \mathbb{Q} as sections of a line bundle (in \mathbb{Z}) over the primes (or spec(\mathbb{Z})), are there other such line bundles? Is there a story in terms of graded commutative rings?

This led Tim Silverman and others to start talking about ‘the real prime’ and ‘archimedean places’ and stuff like that.

It’s fascinating stuff, but I don’t really understand it as well as I’d like, so I just wanted to say that it’s not necessary to understand this stuff to fit number theory into the framework of ‘dimensional theories’ that we’ve been discussing here.

The point is that there’s a pretty well-known thing that completes this analogy:

linebundles:Riemannsurfaces::???:ringsofalgebraicintegers line bundles:Riemann surfaces::???:rings of algebraic integers

and that thing is fractional ideals. These play an incredibly important role in number theory, and they are the objects of a dimensional category which is analogous to the category of line bundles over a Riemann surface.

The idea, very roughly, is that given a ring RR of algebraic integers, like 5\mathbb{Z}\sqrt{-5}, a fractional ideal is a ‘projective RR-module of rank 1’.

You may or may not know what this means, but thanks to the Serre-Swan theorem, when you see ‘projective RR-module’ you should think vector bundle. And when you see ‘projective RR-module of rank 1’, you should think line bundle. So, a fractional ideal acts like a line bundle.

Just as the tensor product of line bundles is a line bundle, the tensor product of fractional ideals is again such a thing. So, we get a symmetric monoidal category of these guys.

And just as every line bundle LL has an inverse L *L^* — a line bundle with LL *L \otimes L^* being isomorphic to the trivial line bundle — every fractional ideal has an inverse. So, we get a dimensional category of these guys!

And if we think of this dimensional category as a GG-graded commutative ring, GG is basically the ideal class group of RR.

So, a bunch of nice traditional stuff fits into the ‘dimensional category’ perspective pretty well, even before we get fancy and start trying to take into account the archimedean valuations.

Posted by: John Baez on July 17, 2009 7:55 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David wrote:

Given that dimensional categories are equivalent to GG-graded commutative rings for some abelian group GG […] are we going to see a generalisation to the noncommutative case?

Jim has a very pretty generalization to the noncommutative/nonabelian case, but I don’t feel like explaining it now. It’s related to ‘Yetter–Drinfel’d doubles’ in a cool way, but I don’t understand its applications to physics in the way you’re suggesting — at least, not yet.

Posted by: John Baez on July 17, 2009 8:17 PM | Permalink | Reply to this
Read the post What is a Theory?
Weblog: The n-Category Café
Excerpt: Is a syntactically presented theory as important as its walking model?
Tracked: July 14, 2010 5:47 AM

Re: Algebraic Geometry for Category Theorists

The mysterious ways of the web led me back to this entry here, and I looked at it’s wiki entry. (Don’t have the time to watch the recorded lectures, though.)

I realize that I still have the same kind of question that I had once voiced above. I’ll formulate it in stronger form,

what is it that is new here?

I am asking this provocatively, but I am genuinely interested in hearing the answer. Possibly it’s all in material in the recorded lectures that is not reflected in the wiki entries that I have seen. But then maybe other readers have the same problem as I do, and would profit from an answer.

I start out feeling a bit irritated by the introductions to the material. Saying that there is no genuine or widely acknowledged general abstract/category theoretic and topos-theoretic basis for algebraic geometry might have been evident a few years back. But it sounds like quite a bold statement today after the work by Kontsevich, Toën-Vezzosi, Lurie, and many others.

The content of Structured Spaces is a vastly general, general abstract foundation not just of algebraic geometry, but of geometry . (It is all based on the notion of (,1)(\infty,1)-algebraic theory, by the way, “(1,1)(\infty-1,1)-doctrines”, if you prefer!) After all, the “algebraic” in “algebraic geometry” is just a choice of site. A general abstract description of it should not crucially depend on making that choice.

More concretely, from your wiki entry I am getting the impression that the central idea you are presenting is to formalize properties of categories of coherent sheaves. And you suggest that the important property is that these form a cocomplete symmetric monoidal linear category.

That sounds good. But it also sounds like something that has been developed to quite some extent. If we go higher up the categorical ladder and consider \infty-categories of quasicoherent \infty-stacks (whose homotopy category is the familiar derived category of quasicoherent sheaves) then we find that

  • linear category becomes stable \infty-category

  • cocomplete category may be taken to be locally presentable \infty-category;.

  • symmetric monoidal cateory becomes symmetric monoidal \infty-category .

These characteristic properties of \infty-categories of quasicoherent \infty-stacks play a central role in the geometric \infty-function theory that we once discussed here. Precisely the fact that you mention on the wiki, that these categories can be regarded as commutative ring objects in higher categories, is what drives the discussion there.

Then I gather from remarks on your wiki site that you intend to think of schemes and stacks and so forth always dually in terms of their cats of quasicoherent sheaves. This point of view has a long tradition. It is the driving force behind Kontsevich’s work over the last decades, I guess: there it goes by the name of noncommutative algebraic geometry and derived noncommutative geometry and stable \infty-categories are modeled as linear A A_\infty-categories, but up to these terminology differences, it seems to me – judging only from the little I saw on your wiki – that this is the kind of setup that you are after.

Could that be? If not, could you maybe highlight again what the guiding idea is that you are following in this program?

Posted by: Urs Schreiber on December 8, 2010 6:51 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Urs wrote:

what is it that is new here?

I don’t really care anymore. Why don’t we just say “nothing” and be done with it?

Posted by: John Baez on December 9, 2010 6:34 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I asked:

what is it that is new here?

John replied:

Why don’t we just say “nothing” and be done with it?

Why don’t we instead say: the basic idea is not new, but let’s stand on the shoulders of giants that came before us and still look a bit further ahead?

It’s sad to drop any subject as soon as it becomes clear that others have had important things to say about it, too.

I would enjoy having more discussion about the general abstract basis of (algebraic) geometry here. It seems that Jim Dolan is still interested in it, too. I am wondering why we can’t have a discussion about it. Is it me? Jim is not responding to me either, here. Maybe I am not being a good discussion partner. That’s too bad (for me).

Posted by: Urs Schreiber on December 9, 2010 1:27 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

what is it that is new here?

it sounds like the kind of “newness” that you’re talking about here refers to the alleged “research frontier”, the frontier of what’s known by the monolithic “research community”. i’ve always considered myself more of a teacher than a researcher and i’m more interested in the “teaching frontier”, the frontier of my own understanding, and of that of the students who i encounter.

anyway, i don’t know which sort of frontier is more worth paying attention to, but perhaps i can perform an experiment. before i read your message here, i mostly finished writing this, where i recommended working out the example of the coherent sheaves over projective n-space as the free symmetric monoidal finitely cocomplete algebroid on a 1-dimensional object linearly embedded in standard linear [n+1]-space. maybe if you could work out this example and post a description of it here then i could get a sense of how the ideas that i’m trying to explain are subsumed by some larger body of ideas that you’re referring to.

i’m very slow at writing so i might have trouble trying to keep up with what you post, but i’ll try to do so.

i hope that i described the example that i’m talking about clearly enough; if not then i can try to explain it more clearly.

Posted by: james dolan on December 10, 2010 1:07 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Jim Dolan wrote had written:

but perhaps i can perform an experiment. […] i recommended working out the example of the coherent sheaves over projective n-space as the free symmetric monoidal finitely cocomplete algebroid on a 1-dimensional object linearly embedded in standard linear [n+1]-space. maybe if you could work out this example and post a description of it here then i could get a sense of how the ideas that i’m trying to explain are subsumed by some larger body of ideas that you’re referring to.

If I understand you correctly, you would like me to tell you how I would suggest to view this situation. In that respect, here is an attempt:

it seems to me that the concept that you are getting at is closely related to that of a perfect stack.

A stack is called perfect if its compact quasicoherent sheaves are precisely the dualizable quasicoherent sheaves and if the whole category of quasicoherent sheaves is generated from these compact/dualizable objects.

(In particular affine, projective and generally quasi-projective schemes (and \infty-schemes) are perfect stacks (\infty-stacks).)

Could that be?

Posted by: Urs Schreiber on December 13, 2010 6:00 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i’m not sure i understand how what you wrote is supposed to be responding to my suggestion. i’ll try clarifying my suggestion.

i’m asking you to prove (or sketch a proof of) a certain theorem. my previous statement of the theorem was clumsy (or maybe not quite correct) so i’ll try to state it more clearly here.

(it could also help if someone can give a reference to a proof of the theorem or of some stronger theorem; or if the theorem is false in some way then explain how.)

theorem: the coherent sheaves over projective n-space form the free symmetric monoidal finitely cocomplete algebroid on a line object l equipped with an epimorphism e : n+1 -> l. (a “line object” here is an object invertible wrt tensor product and having trivial self-braiding, and “n” is the direct sum of n copies of the unit object 1. for now i’ll restrict to the case where we’re working over the complex numbers.)

some of what you wrote seemed to be going partway towards sketching some sort of proof of this theorem, but only partway as far as i was able to make out.

my suggestion for you to try to prove this theorem is based on the idea that you might be able to show me how it is subsumed under some larger body of ideas, or alternatively that you could get some idea of how the viewpoint that i’m taking here compares to other viewpoints.


Posted by: james dolan on December 16, 2010 1:33 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I’m probably missing something (everything?) since I am unfamiliar with much of the categorical language being used (doctrines, algebroids, even topoi to an extent) but I believe what you’re asserting is a special case of Tannakian reconstruction, in the form e.g. presented by Jacob Lurie in his second arxiv posting (to which Urs refers above).

To paraphrase: to any symmetric monoidal category C we assign a functor Spec C from rings to groupoids, by setting Spec C(k)=tensor functors from C to k-modules. A form of Tannakian reconstruction says that if C is quasicoherent sheaves on a variety or geometric stack, then Spec C recovers the variety - ie we get exactly the functor of points on the variety (in this case we get a functor to discrete groupoids, ie sets).

So now we look at C=quasicoherent sheaves on P^n (I put in quasi to be closed under infinite direct sums, which seems to match what you’re looking for). Then we recover Spec C(k) = k-points of P^n, in other words lines in an n+1 dimensional vector space. Is this not precisely the assertion that C is the free symmetric monoidal category on a line object in n+1 space?

(There’s a similar assertion on the \infty-categorical level, ie for derived categories, but there we really should restrict to varieties or something close to avoid annoying counterexamples..)

Posted by: David Ben-Zvi on December 16, 2010 2:59 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Perhaps the difference between our assertions is that I’m specifying the Homs from the tensor category to all k-Mod, while you’re specifying its Homs to all other tensor categories (in other words perhaps you are specifying a priori what “a line in an n+1 dim space” means over any tensor category, while I’m just using its usual definition over rings). I understand the appeal of this more categorical assertion. However it follows from the Tannakian story that some large class of tensor categories is generated by k-Mod’s (ie specifying Homs to k-Mod’s determines the tensor category). Unfortunately as Jacob complains there doesn’t seem to be a good way to identify the essential image of geometric stacks in the world of all tensor categories..

Posted by: David Ben-Zvi on December 16, 2010 3:37 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Perhaps the difference between our assertions is that I’m specifying the Homs from the tensor category to all k-Mod, while you’re specifying its Homs to all other tensor categories (in other words perhaps you are specifying a priori what “a line in an n+1 dim space” means over any tensor category, while I’m just using its usual definition over rings).

i think that you’re right that something very much like that is the difference between the assertions, though i have to think some more about what you and urs and jacob are saying to see how well i understand all that’s going on here.

i think that the extra that i’m asking for is fairly easy to supply by direct means, and i’ll try to describe that at some point. but it would be interesting if it can also be supplied by some other means such as you guys might be hinting at.

Posted by: james dolan on December 16, 2010 4:50 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Great! looking forward to having things sorted out. Let me just add a comment that might help interpret the Tannakian story: the functor Spec from tensor categories to stacks is really just an adjoint to the functor QC sending a stack to its tensor category of quasicoherent sheaves. In this language Tannaka reconstruction is the assertion that the two functors are equivalences on the essential image of geometric stacks by QC. That helps me at least to think about Spec.

Posted by: David Ben-Zvi on December 16, 2010 5:54 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

let me give a short recap before trying to describe the idea that i have in mind.

the question that we’re dealing with is essentially: to what extent (or in what context) is it true that projective n-space is the classifying object for “a line bundle equipped with n+1 sections that generate it”?

it’s true essentially by definition for bundles over an affine scheme, and thus also for locally trivial bundles over any stack covered by affine pieces. so it’s true in a very general context; which is a good thing, because this classifying (or “universal”) property is the whole point of what projective n-space is supposed to be.

furthermore, the “tannakian” correspondence between stacks and something like “tensor categories” gives some hope of establishing the classifying property for line bundles (or rather, in this more abstract setting, “line objects”) in any tensor category, but the imperfect nature of the correspondence complicates the task of actually doing this.

instead, we can establish the classifying property for line objects in any tensor category by a direct method, bypassing the use of the tannakian correspondence, if “tensor category” is now taken to mean “symmetric monoidal finitely cocomplete algebroid”. this direct method, which i’ll try to describe now, is at the very least implicit in the usual ways that algebraic geometers think about line bundles and projective embeddings and so forth, but presumably it’s also explicitly spelled out somewhere.

first, consider the “homogeneous coordinate algebra” of the usual (“tautological”) projective embedding of projective n-space; this is the z-graded commutative algebra of polynomials in n+1 variables (all of which live in grade 1).

under the correspondence between graded commutative algebras and “dimensional categories”, this corresponds to the free dimensional category on one object l equipped with a “virtual” morphism e from the direct sum of n+1 copies of the unit object. (“virtual” here refers to the fact that although actual direct sums of objects don’t exist in a dimensional category, morphisms between formal direct sums can be encoded as matrixes of morphisms.) that is, the universal property of the polynomial algebra in the 1-category of z-graded commutative algebras translates under this correspondence into the stated universal property in the 2-category of dimensional categories.

but now we can freely adjoin finite colimits to the dimensional category, in particular promoting the formal direct sums to actual direct sums. thinking of a dimensional category as an algebroid, the usual way to freely adjoin finite colimits to an algebroid is to take its finitely presented contravariant modules. when the algebroid possesses a tensor product t, “day convolution” of the modules wrt t is the co-continuous (or “half-exact”) extension of t. in this case, the contravariant modules of the dimensional category are just the z-graded modules of the original z-graded commutative algebra, and day convolution is the usual “graded tensor product” of graded modules. the original objects of the dimensional category become line objects in the category of graded modules.

thus, the finitely presented graded modules of the homogeneous coordinate algebra of projective n-space form the free symmetric monoidal finitely cocomplete algebroid on a line object l equipped with a morphism e from the direct sum of n+1 copies of the unit object.

at this point, we can recall (or learn for the first time, as the case may be) that there’s a functor called “proj” that takes finitely presented graded modules of the homogeneous coordinate algebra of a projective variety x to coherent sheaves over x. (after all, we should expect some way to understand coherent sheaves over x in terms of its homogeneous coordinate algebra, and this is it.) it’s easy to show that this functor “proj” is the universal way to force the morphism e to be an epimorphism while preserving tensor products and finite colimits, and actually doing so then completes the proof that the coherent sheaves over projective n-space form the free symmetric monoidal finitely cocomplete algebroid on a line object l equipped with an epimorphism e from the direct sum of n+1 copies of the unit object.

this theorem suggests the idea of using this sort of characterization of coherent sheaves (over geometrically interesting stacks such as projective n-space) as the primary definition of what coherent sheaves are, and of what the stacks over which they live are, bypassing other definitions that rely on the formalism of stacks and sheaves and so forth. this is still in the spirit of the tannakian philosophy of the correspondence between the “algebraic” and the “geometric” viewpoints, perhaps even more so, but exploiting that philosophy in a somewhat different way that (following tom leinster’s thoughts) seems particularly suited for category theorists to understand, but which also seems relatively unknown to them so far, in my experience.

alex hoffnung and i have tried to develop many other examples of such characterizations of coherent sheaves over geometrically interesting stacks. for example, the rational representations of the upper-triangular subgroup of the algebraic group gl(3), aka “coherent sheaves over the stack of flagged 3-dimensional vector spaces”, form the free symmetric monoidal finitely cocomplete algebroid on a flagged 3-dimensional object, where however it’s important to formulate the concept of “flagged 3-dimensional object” using only the syntax of symmetric monoidal finitely cocomplete algebroids (basically tensor products and finite colimits). we can give a similar characterization of the coherent sheaves over the stack of “bi-flagged 3-dimensional vector spaces” and use it in the study of categorification of hecke algebras, for example. we can also generalize these ideas to the geometry of an algebraic group associated with an arbitrary dynkin diagram.

in some sense, dynkin diagrams themselves are another syntax for writing down theories describing geometric objects of a certain kind, and alex and i are trying to systematize the process of translating from the syntax of dynkin diagrams to the syntax of symmetric monoidal finitely cocomplete algebroids. coxeter diagrams are another such syntax, extractable from dynkin diagrams. the geometric objects associated with coxeter diagrams are “incidence geometries” or so-called “buildings”, which are extractable from the geometric objects associated with dynkin diagrams. todd trimble and i have tried to systematize the process of translating from the syntax of coxeter diagrams to the syntax of coherent toposes, and alex and i have begun to think about the relationship between the algebraic stacks extracted from dynkin diagrams and the coherent toposes extracted from their underlying coxeter diagrams.

of course there’s still a lot of unanswered questions here and mistakes to be fixed, and we need to try to understand how what we’re trying to do relates to what others are doing.

Posted by: james dolan on December 22, 2010 6:45 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

In addition to the graded case/projective varieties one can in complete analogy treat the multigraded algebras (algebras graded over finitely generated free abelian groups) and the refined Proj (as well as Cone) over the corresponding lattice of higher rank. For a generalization in noncommutative direction one can compare with the approach of

  • M. Artin, J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), no. 2, 228-287. MR96a:14004, doi

It is interesting that you are getting in the analysis of examples from your perspective to the combinatorics flag manifolds and alike spaces, which are known to be typical blocks not only of traditional classifying spaces, but of interesting moduli spaces in geometry in general (there was an interesting long-term project on “applied homotopy theory” by Millman focusing precisely on that aspect). My understanding is that you are investigating the corresponding syntax in more detail than it is usually done, what must be rewarding.

Posted by: Zoran Skoda on December 23, 2010 12:18 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i have a naive question about a sort of algebraic stack that seems “geometrically” natural and non-pathological to me, but whose place in any sort of “tannakian correspondence” seems somewhat problematic.

the sort of example that i’m thinking about here is the classifying stack of a non-affine algebraic group (such as an abelian variety), or more generally any sort of stack where some isotropy groups are non-affine. i would expect a shortage of “coherent sheaves” over such a stack, because non-affine groups seem to have a shortage of “algebraic” representations.

so what sort of tannakian correspondence, if any, do such stacks fit into? i’ve thought a bit about trying to fit them into some sort of “higher” such correspondence, in various ways, but not very conclusively so far.

it almost seems like there should be a progression of higher tannakian correspondences of some sort, as the simplest flavor of tannaka-krein reconstruction automatically reconstructs an affine (or maybe “pro-affine”?) group from a sufficiently good tensor category, whereas lots of non-affine schemes qualify as “2-affine” in the sense of fitting into the more general tannakian correspondence. of course there may be various concepts of “n-stack” but here i’m trying to imagine instead some different though perhaps related concept of “n-affine”. i’m not sure how much sense this actually makes though.

(i should mention that i’m way behind on my reading, in case this is discussed in some of the literature that’s been mentioned here.)

Posted by: james dolan on December 23, 2010 2:51 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

James – that’s a question that I’m very fond on and that David Nadler and I have thought through to some extent. As you point out, affine schemes are affine for functions O. Stacks with affine diagonal (“geometric stacks” - might want to assume quasicompactness too) are affine for quasicoherent sheaves, i.e. O-modules. You can define an n-affine stack as one whose diagonal morphism is (n-1)-affine, and with the right categorical machinery (higher version of Barr-Beck) it should be easy to see that these are affine for quasicoherent (((O-mod)..-mod)-mod.

BTW there are two different ways to think about this affineness. The Tannakian question asks to reconstruct an affine scheme (geometric stack) as Spec of its assigned commutative ring (symmetric monoidal category). But you can also interpret affineness as the statement that for an affine scheme global sections induces an equivalence between quasicoherent sheaves and modules for this ring. Likewise 2-affineness means that there’s an equivalence between the 2-categories of module categories for the symmetric monoidal category QC(X) (X geometric stack) and of quasicoherent sheaves of categories over X. The latter picture is quite useful, and John Francis, Nadler and I use it to establish a 2-Morita equivalence between “Hecke categories” (monoidal categories arising as “matrix algebras” over a base” and the base.

I should say that our motivation for considering say 3-affine stacks comes from the world of D-modules. D-modules on a variety or stack X can be identified with quasicoherent sheaves on the de Rham space X{dR} introduced by Simpson (the quotient of X by the formal neighborhood of the diagonal). However X{dR} basically never has affine diagonal (even for X= the affine line!) So none of the usual Tannakian stories apply to D-modules — that’s the basic issue behind Nadler and my paper “Character theory..”, though we don’t talk about it from this point of view much.

Posted by: David Ben-Zvi on December 23, 2010 3:41 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i just want to try to see how much i’m understanding of what you’re saying here…

so this “de rham space” that you’re talking about is a non-geometric stack? and the quasicoherent sheaves on it are the d-modules, which form a reasonably nice tensor category, in particular an abelian category with tensor product and internal hom related via hom-tensor adjointness? and the “spectrum” of this tensor category is another (still non-geometric??) stack, with the comparison morphism from the de rham space being a non-equivalence?

i didn’t quite work out yet how any nontrivial example of a 3-affine stack might be arising here…

from my viewpoint, what seems most interesting here (so far) is how the d-modules have great usefulness and a nice “geometric interpretation” and (what to me is almost the same thing in this context) a nice universal property as a symmetric monoidal cocomplete algebroid despite apparently not fitting into the tannakian correspondence as ordinarily construed.


Posted by: james dolan on December 24, 2010 3:31 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I was a little hasty/imprecise about D-modules above. The main technical problem with D-modules from our point of view is that pushforward is essentially never conservative (except for finite maps) - that’s the sense in which D-modules behave non-affinely even on affine varieties..and which is why on stacks with affine diagonal Tannakian constructions with D-modules – eg description of sheaves on a fiber product as a categorical tensor product of categories of sheaves – fail dramatically.. though they DO hold for schemes.

In fact I never asked myself this question before but it seems to me that Tannakian reconstruction DOES work for D-modules on schemes X.. there’s a natural map from X to Spec D(X) given by the forgetful functor on D-modules to O-modules (ie pullback from the de Rham space to X). Moreover if two points are infinitesimally close the corresponding fiber functors on D-modules will be identified. So this map descends to the de Rham space of X, where I would guess it’s an isomorphism. I guess you have to be carefuller than I was in claiming Spec C has affine diagonal (for D-modules it’s pretty close to affine - the de Rham space is a quotient of X by an ind-affine equivalence relation, so not so far..)

In any case I have yet to be convinced that there’s something “beyond Tannakian” in the universal properties for tensor categories — ie the question is to what extent tensor categories of the form k-mod generate all tensor categories. If you could give me a reasonable tensor category with no nontrivial Homs to modules over a commutative ring I’d be convinced.

Posted by: David Ben-Zvi on December 24, 2010 4:47 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I got very helpful suggestions from Noah Snyder and Victor Ostrik. In particular Deligne’s categories Rep(GLt), Rep(Ot), Rep(St) apparently do not have tensor functors to any category of the form R-mod (for references Victor suggested an entry of David Speyer on the Secret Blogging Seminar, a paper and blog entry by Akhil Matthews, and a paper of his and Jonny Comes). So this suggests indeed the importance of non-Tannakian approaches to characterizing tensor categories!

Posted by: David Ben-Zvi on December 24, 2010 8:48 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

In addition to S_t, GL_t, and OSP_t, there’s one other example I know of a non-Tannakian symmetric tensor category (which I learned about from Pavel Etingof). You start with the semisimplified representation theory of U_q(sl_2) at q a pth root of unity and then you reduce modulo p (you need to take a little bit of care to make this reduction make sense). This reduction modulo p makes the usual braided structure actually symmetric. This should generalize to some other quantum group categories modulo p.

Posted by: Noah Snyder on December 24, 2010 10:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Could there not exist something like “sets with t elements” with some “St” as automorphisms which specialize to usual sets and Sn, and “fractional vector spaces” on which “St” operates (and maps into “Glt”)?

Posted by: Thomas on December 25, 2010 7:09 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i suspect that i have too many questions now to be able to efficiently write them down, so i don’t know how much more i might be able to contribute to the discussion for now.

in the course of the discussion i did however realize that i’ve been thinking about d-modules the wrong way for a couple of decades, and that it should be a lot easier to understand them now that i think that i’ve straightened that out; so i’m glad that that happened. but i still need to think some more to try to understand what sort of “non-affine” behavior of d-modules you’re talking about.

anyway, if we get the chance to talk sometime there’s a lot of questions that i’d probably like to ask you.

Posted by: james dolan on December 26, 2010 5:20 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

In any case I have yet to be convinced that there’s something “beyond Tannakian” in the universal properties for tensor categories — ie the question is to what extent tensor categories of the form k-mod generate all tensor categories. If you could give me a reasonable tensor category with no nontrivial Homs to modules over a commutative ring I’d be convinced.

well, i’m not sure how close what i’m suggesting is to what you’re suggesting i’m suggesting (particularly since i seem to have misunderstood some of what you were trying to say about various concepts of “affine”); tensor categories without realizations in module categories probably isn’t particularly a motivation of mine.

one question that i’m interested in though is the completeness and cocompleteness properties (in the (2,1) sense) of various (2,1)-categories. the concept of “doctrine” that i’m interested in is essentially equivalent to that of “locally presentable (2,1)-category”. the “tensor categories” that i’m mainly interested in, namely the symmetric monoidal finitely cocomplete algebroids, manifestly form a locally presentable (2,1)-category (being the models of a categorified limit-sketch), but i’m unsure as to whether other versions of “the (2,1)-category of (symmetric) tensor categories” are locally presentable.

thus although we’ve been talking about “tensor categories”, i’m not sure of the precise meaning of that term, or even whether any very precise meaning is intended. (for example the n-lab entry on “tensor category” leaves it vague.) on the other hand, in the context of studying some sort of tannakian correspondence as an adjunction of (2,1)-functors, we should be more precise in specifying the (2,1)-category of “tensor categories” as one side of that adjunction; so could someone say here just what that (2,1)-category is, and whether it’s known to be locally presentable? (i guess that the references that people have already given me should answer that, but some of those references are pretty long, so perhaps someone can narrow it down a bit.)

Posted by: james dolan on December 26, 2010 10:19 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

i guess that the references that people have already given me should answer that, but some of those references are pretty long, so perhaps someone can narrow it down a bit.

I tried to write a concise summary with the precise definitions and statements at Tannaka duality for geometric stacks.

For instance concerning this point here:

thus although we’ve been talking about “tensor categories”, i’m not sure of the precise meaning of that term, or even whether any very precise meaning is intended.

The definition of TensorCategoriesTensorCategories that makes forming quasicoherent sheaves

QC:GeometricStacksTensorCategories op QC : GeometricStacks \to TensorCategories^{op}

a full and faithful (2,1)(2,1)-functor is the following:

An object is a symmetric monoidal category (C,)(C, \otimes) whose underlying category CC is abelian and satisfies the AB5-axion, such that for all cCc\in C tensoring ()c:CC(-) \otimes c: C \to C commutes with all colimits and preserves all short exact sequences whose third object is flat.

A morphism is a symmetric monoidal functor that commutes with all small colimits and preserves flat objects and short exact sequences whose third object is flat.

A 2-morphism is a natural isomorphism between such functors.

(So this is a bit technical and I am not sure how to see whether this makes TensorCategoriesTensorCategories be locally presentable.)

Posted by: Urs Schreiber on December 28, 2010 3:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I don’t know the tensor category literature well but in the \infty-categorical setting local presentability is discussed e.g. in Toen-Vezzosi’s arXiv:3903.3292..

Posted by: David Ben-Zvi on December 28, 2010 8:33 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

that was more futuristic than intended.. I meant 0903.3292..

Posted by: David Ben-Zvi on December 28, 2010 8:35 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

[…] affine schemes are affine for functions O. Stacks with affine diagonal (“geometric stacks” […]) are affine for quasicoherent sheaves, i.e. O-modules. You can define an n-affine stack as one whose diagonal morphism is (n-1)-affine, […]

It is maybe interesting that there is, as you know, another notion of higher stacks that generalizes the first two of these examples:

the notion of geometric \infty-stacks: realizations of \infty-groupoid objects in \infty-stacks that are degreewise in the image of the (,1)(\infty,1)-Spec functor Spec:TAlg opSh (TAlg op)Spec : T Alg_\infty^{op} \to Sh_\infty(T Alg^{op}).

Do you know how these “\infty-groupoids in 1-affine \infty-stacks” relate (or should relate) generally, for n2n \geq 2 to nn-affine \infty-stacks? (For TT the theory of ordinary kk-algebras or more generally.)

Posted by: Urs Schreiber on December 23, 2010 8:55 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Hi Urs – the notion of n-affineness as I discuss above is if I’m not mistaken a filtration on the collection of all [quasicompact?] algebraic higher stacks, which is I believe what you’re discussing (modulo some boundedness issues)? In other words, you are discussing stacks equivalent to simplicial affine schemes, in other words a stack with an affine covering and algebraic relations. I’m asking those relations to be either 1. trivial, 2. affine, 3. 2-affine etc. (sorry not thinking very clearly lately..)

Posted by: David Ben-Zvi on December 24, 2010 12:56 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Hi Urs – the notion of n-affineness as I discuss above is if I’m not mistaken a filtration on the collection of all [quasicompact?] algebraic higher stacks, which is I believe what you’re discussing (modulo some boundedness issues)?

What I had been wondering about was the relation of (n+1)(n+1)-affine stacks to geometric nn-stacks (in the sense of Toën) for n2n \geq 2.

A geometric 0-stack is an affine variety XX. This is equivalent to the 1-Spec of its 1-algebra of functions 𝒪(X)\mathcal{O}(X).

A geometric (1-)stack XX is equivalent to the 2-Spec of its 2-algebra quasicoherent sheaves QC(X)QC(X), which is locally on 1-affine UXU \to X given by QC(U)=𝒪(U)ModQC(U) = \mathcal{O}(U) Mod.

This might be the beginning of a pattern: say that a geometric nn-stack is an nn-truncated geometric \infty-stack. I am wondering: is it then true (in some suitable sense) that a geometric nn-stack is the (n+1)(n+1)-Spec of its (n+1)(n+1)-algebra of (n+1)(n+1)-functions that are locally on affine UXU \to X given by 𝒪(U)ModMod\mathcal{O}(U) Mod \cdots Mod?

In another message (here) you mention how the de Rham stack X dRX_{dR} of an affine scheme XX fails to be 2-affine. There is however the de Rham schematic homotopy type (X dR,x)(X^{dR},x) for every connected comoponent of XX, and this is a geometric \infty-stack. So it should be useful to understand the relation between (n+1)(n+1)-affine stacks and geometric nn-stacks for n>1n \gt 1.

Posted by: Urs Schreiber on December 28, 2010 6:13 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I think we (following Lurie and Toen) are using geometric stacks in two separate senses — the ones which are 2-affine are those geometric stacks in the sense you mean which in addition have affine diagonal.. which is more restrictive.. e.g. the moduli of genus one curves is not geometric in this sense (since each has an elliptic curve as automorphisms). I proposed a sense of higher affine stacks for which I think your question about n+1-spec holds, but it certainly doesn’t for just n-truncated Artin stacks.

Posted by: David Ben-Zvi on December 28, 2010 8:04 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I think we (following Lurie and Toen) are using geometric stacks in two separate senses

Okay, please let’s briefly gauge our terminology then. I am fine with using whatever terminology you suggest, but right now it seems I am not exactly sure which one it is.

I proposed a sense of higher affine stacks

Just to be sure: this is the definition that an \infty-stack is nn-affine if its diagonal is (n1)(n-1)-affine? If not, could you do me the favor and just state the definitin explicitly again? Thanks.

(Can we keep around the “\infty” where relevant for the time being? It seems to matter in a discussion where “stack” also used to implicitly mean “1-truncated \infty-stack” and where the truncation degree matters crucially.)

it certainly doesn’t for just n-truncated Artin stacks.

Again just so I am sure that I understand your terminology: are you referring to the “geometric \infty-stacks” from Toën’s Champs affine as just Artin stacks here? A priori I wouldn’t think that’s what you’d do, but if you don’t, I can’t see how your remark is a reply to what I said, sorry. I keep referring to what Toën calls champs \infty-géométriques . Please let me know what you call these.

Posted by: Urs Schreiber on December 28, 2010 9:06 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

OK, so we’re getting confused between THREE notions of stack in our conversation, not just two!
I’ll refrain from being too precise so as to attempt not to commit myself to more public stupidities than I already have.. but let’s see: In increasing order of generality, there’s what I was calling n-affine above (which is probably terrible terminology and maybe not a useful notion – and not to be confused with Toen’s affine stacks, though there’s a relation one way). At the first stage, these are affine schemes glued by affine relations (eg an affine scheme mod an affine group scheme). Then there are higher Artin or algebraic stacks, which at the first stage are affine schemes glued together by smooth relations (eg an affine scheme mod any group scheme in char=0). Then there’s the notion Toen introduces in that paper (don’t know if it’s very common terminology though?) as geometric stacks, which at the first stage are affine STACKS in Toen’s sense modulo smooth relations.. eg spec of any simplicial commutative algebra mod a group scheme. And then one proceeds in an iterative way. Hope that helps..

Posted by: David Ben-Zvi on December 29, 2010 4:07 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Then there’s the notion Toen introduces in that paper (don’t know if it’s very common terminology though?) as geometric stacks

I have seen this terminology only in his articles. But also, I have seen nobody else say “geometric \infty-stack” at all, so relatively speaking, it is still seems to be very common ;-)

And, yes, as I tried to say above, Toëns “geometric \infty-stacks” are groupoids internal to \infty-stacks in the image of Spec:(Alg Δ) opStacksSpec : (Alg^\Delta)^{op} \to \infty Stacks.

What makes this interesting (to me at least) is that this is a nicely intrinsic notion (which is not to that extent true for the other definitions, it seems to me).

Namely Toën shows in his article (somewhat in between the lines of his model category theoretic treatment) that the image of Spec:(Alg 𝒰 Δ) opStacksSpec : (Alg_{\mathcal{U}}^\Delta)^{op} \to \infty Stacks is precisely (up to a certain size issue that can be dealt with) the cohomology localization of the full \infty-topos at cohomology with coefficients in the additive group 𝔾 a\mathbb{G}_a.

So his “affine stacks” are precisely the (small) H (,𝔾 a)H^\bullet(-,\mathbb{G}_a)-local objects.

And his “geometric \infty-stacks” are groupoid objects in these H (,𝔾 a)H^\bullet(-, \mathbb{G}_a)-local objects.

I got interested in this because the analogous statement is true with ordinary algebras generalized to algebras over any abelian algebraic theory, under some mild conditions, with 𝔾 a\mathbb{G}_a replaced by the corresponding canonical line object and its abelian group structure. In particular this makes sense not only for algebraic but also for topological and differentiable stacks.

I also keep wondering how his constructions lift to the derived case. In your article with David Nadler you replace his (𝒪Spec)(\mathcal{O} \dashv Spec)-adjunction by the \infty-Kan extension of dgAlg dgAlgdgAlg^- \hookrightarrow dgAlg along the \infty-Yoneda embedding. But, as you remark Toën observed, this Kan extension fails to yield a reflective embedding, so what you consider cannot be the derived cohomology localization. I would like to know if the cohomology localization of the derived topos exists, what it is, and how it relates to Toën’s underived \infty-construction.

Posted by: Urs Schreiber on December 29, 2010 5:40 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

while i’m still trying to understand david’s answer to my last one, i have another naive question about (roughly) the relationship between algebraic stacks and tensor categories.

consider the contravariant (2,1)-functor that assigns to a symmetric monoidal finitely cocomplete k-algebroid x the pre-stack on the category of affine k-schemes given by homming x into the modules over the coordinate k-algebra of the affine k-scheme.

what is the grothendieck topology with as many coverings as possible for which all of the pre-stacks in the image of this (2,1)-functor are stacks? how does this topology relate to other known grothendieck topologies on the same site category?

of course i’d also be interested in suggestions about how to fine-tune the question.

Posted by: james dolan on December 24, 2010 2:32 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

James – I don’t know the finest topology in which Spec C is a stack for a tensor category, but the flat topologies work (there are two commonly studied versions, in which we require coverings to be faithfully flat and either finite presentation (fppf) or quasicompact (fpqc)). The reason is that descent for quasicoherent sheaves works in the flat topologies (thanks to Grothendieck) - in fact the flatness seems rather close to the Barr-Beck conditions for descent to hold (certainly faithful [=> conservative pullback] and flatness [=>exact pullback] seem necessary) so they’re probably close to optimal (I’m sure mathoverflow will provide a precise answer if asked!)

The argument for Spec C to form a stack in a topology satisfying descent for sheaves (which I learned from John Francis) is simple: functors out of C commute with limits in the target, and if we have a flat cover of X we can write sheaves on X as a limit of sheaves on the assciated Cech simplicial scheme (this argument works on the derived, ie \infty-categorical, context just as well).

Posted by: David Ben-Zvi on December 24, 2010 3:48 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

David Ben-Zvi says to James Dolan:

I believe what you’re asserting is a special case of Tannakian reconstruction, in the form e.g. presented by Jacob Lurie in his second arxiv posting (to which Urs refers above).

Since I mentioned this in the other thread, for the record: this refers to theorem 5.11 in Tannaka duality for geometric stacks.

David also writes:

Let me just add a comment that might help interpret the Tannakian story: the functor Spec from tensor categories to stacks is really just an adjoint to the functor QC sending a stack to its tensor category of quasicoherent sheaves. In this language Tannaka reconstruction is the assertion that the two functors are equivalences on the essential image of geometric stacks by QC. That helps me at least to think about Spec.

This is the point that I tried, maybe unsuccessfully, to make in this discussion here: that the proposal promoted here may be understood as thinking of algebraic geometric objects XX not as (locally) formal duals to commutative 1-algebras of functions 𝒪(X)\mathcal{O}(X), but to formal duals of “commutative 2-algebras of 2-functions” (symmetric monoidal abelian categories) QC(X)QC(X).

We have the usual adjunction (“Isbell duality”)

(𝒪Spec):Algebras opSpec𝒪Spaces (\mathcal{O} \dashv Spec) : Algebras^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} Spaces

where SpecSpec regards an algebra (over some Lawvere theory in general, or ordinary algebra in particular) as the formal dual to certain spaces/stacks.

If instead we think of a stack XX as being the formal dual to its monoidal abelian category of quasicoherent sheaves QC(X)QC(X), then this is boosted up to

(QC()2Spec):2Algebras op2SpecQC()Spaces (QC(-) \dashv 2Spec) : 2Algebras^{op} \stackrel{\overset{QC(-)}{\leftarrow}}{\underset{2Spec}{\to}} Spaces

where “2-algebras” here means some flavor of “symmetric monoidal abelian cateories”: commutative algebra objects in the 2-category of abelian 2-categories.

James Dolan writes to me:

i’m not sure i understand how what you wrote is supposed to be responding to my suggestion.

I had taken the liberty of responding only to the second part of your suggestion. Your suggestion had been:

maybe if you could work out this example and post a description of it here then i could get a sense of how the ideas that i’m trying to explain are subsumed by some larger body of ideas that you’re referring to.

I didn’t yet hand in the exercise sheet with a worked proof of the characterization of QC( k)QC(\mathbb{P}^k) (roughly guided exercise 7 here, worth two credit points, I suppose). Instead I jumped to the second part of the suggestion and tried to give a sense of how these observations fit into a larger body of ideas:

I was suggesting that the relevance or meaning of having QC(X)QC(X) be nicely generated from some of its objects is that this means that XX is a perfect stack , which in turn means that on such XX regarding the assignment QC():XQC(X)QC(-) : X \mapsto QC(X) as assigning “2-algebras of 2-functions” makes particularly good sense, in that we have that then QC()QC(-) provides a geometric \infty-function theory.

I am thinking this aspect should be of interest to you, because it is a variant of your notion (or the version of it that John has exposed) of “groupoidification”.

Your groupoidification is (as far as John has exposed it and as far as I understand that) the geometric \infty-function theory of plain \infty-sheaves in the sense that is discussed at integral transforms on sheaves.

This \infty-function theory always exists in every \infty-topos, but it is somewhat “non-linear” in a sense. If however we look at an \infty-topos over formal duals of algebras and then restrict attention to the “perfect” objects in there, also the assignment of \infty-sheaves of modules over the structure ring, QC()QC(-), provides a geometric \infty-function theory.

It’s such kind of statements that I was looking for when I restarted this discussion here. But I feel sufficiently happy now with my understanding of this context of your proposal that I am interested in looking into concrete exercises, computations and proofs – if we write them out on the nnLab…

Posted by: Urs Schreiber on December 16, 2010 7:43 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Urs wrote:

Maybe I am not being a good discussion partner.

Not for me, anyway.

One thing I really liked about Jim’s ideas was that they provide a nice new way to learn the classical concepts of algebraic geometry, such as projective varieties, graded algebras, ample line bundles, the Segre embedding, etcetera etcetera. I never really enjoyed these concepts or understood why they were ‘inevitable’ until he explained them in his own way.

I think an introductory textbook on algebraic geometry could be written that takes this point of view. It could be quite readable and fun. But it doesn’t exist yet. That’s why I said:

But alas, introductions to topos theory don’t seem to explain much about algebraic geometry, and introductions to algebraic geometry don’t seem to fully embrace topos theory. It seems that Grothendieck’s revolution never fully caught on. And that’s sort of sad.

Somehow you translated this into:

Saying that there is no genuine or widely acknowledged general abstract/category theoretic and topos-theoretic basis for algebraic geometry…

which is very different.

You don’t seem very interested in traditional algebraic geometry, or the problems that beginners like me face in learning that subject. You seem more interested in generalizations of algebraic geometry, and the cutting-edge work of experts. For you,

the “algebraic” in “algebraic geometry” is just a choice of site.

and you’re happy that:

The content of Structured Spaces is a vastly general, general abstract foundation not just of algebraic geometry, but of geometry.

That’s fine, of course. But that’s not what I was interested in, back when I wrote this stuff. I was trying to learn, not vast general abstractions, but algebraic geometry.

And now I’m not even trying to do that! I still enjoy it when Jim tells me what he’s doing, but these days I’m trying to switch over to working on environmental issues, energy policy and applied math. So, any sort of conversation where I feel you’re slamming me up against a wall and demanding that I ‘defend’ old comments of mine — I have zero interest in that.

I’m glad to see that you and Jim are starting a conversation, though.

Posted by: John Baez on December 10, 2010 3:03 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

You don’t seem very interested in traditional algebraic geometry,

There is a misunderstanding here:

In which way is it that topos theory/category theory is a means to understand traditional algebraic geometry? Ever since Grothendieck’s revolution that you mentioned it is this:

there is general abstract geometry inside every topos. All concepts are well motivated here (for a category theorist, at least). Then choose the particular topos of sheaves on the étale site, and all these general abstract notions turn into the concrete notions of traditional algebraic geometry.

For instance you mention graded algebras. How does topos theory explain graded algebras? Like this:

Every topos over the site of formal duals of algebras over any Lawvere theory canonically contains an object 𝔸 1\mathbb{A}^1 – the canonical line object. If the Lawvere theory contains the theory of groups, this has underlying it a group object, called 𝔾\mathbb{G}. So it makes sense to consider objects XX in the topos that are equipped with an action 𝔾×XX\mathbb{G} \times X \to X of this group object.

Now implementing this general abstract setup in the topos of traditional algebraic geometry, one finds that the representable objects with such an action of the canonical group object are the formal duals to graded algebras. The grading is the concrete specific incarnation of the general abstract 𝔾\mathbb{G}-action.

I think Grothendieck’s revolution has been to see that traditional algebraic geometry is general abstract topos theory applied to a particular topos. For instance traditional Galois theory of schemes is just Galois theory of toposes applied to the étale topos (called “Grothendieck’s Galois theory”!), as we are just discussing over in another thread.

So therefore I thought that when you started to explain algebraic geometry to categeory theorists, you were trying to pursue this program. And so I pointed out that since Grothendieck, several people have considerably pushed his revolution even a bit further, with formalizing even more of the specific concrete in terms of the general abstract that category theorists like.

For instance you could ask: why is it that traditional algebraic geometry cares so much about the SpecSpec, and about the notion of scheme (kind of the defining notion of traditional algebraic geometry). How could a category theorist understand where these concepts come from? The answer to this is in Lurie’s Structured Spaces . It derives this from some general abstract topos theory and universal algebra.

Posted by: Urs Schreiber on December 10, 2010 9:15 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

So therefore I thought that when you started to explain algebraic geometry to category theorists, you were trying to pursue this program.

I was actually trying to explain Jim’s idea of treating a projective algebraic variety as a theory in a certain doctrine, so that its points are models. This is what is described on the page this blog entry is a pointer to, and this is also what I was discussing with David Corfield here in the comments to this blog entry.

This particular idea is just a small part of Jim’s story, but it makes a good starting-point. This particular starting-point does not require that one introduce concepts such as ‘scheme’, ‘sheaf’, ‘site’ or ‘topos’. You can think of it as a way to teach ‘classical’ algebraic geometry — the geometry of projective varieties — with the help of ideas from category theory.

Of course, topos theory comes in naturally at some later point in the story… but that’s not actually what I was most interested in.

Posted by: John Baez on December 10, 2010 10:02 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I was actually trying to explain Jim’s idea of treating a projective algebraic variety as a theory in a certain doctrine, so that its points are models.

This is asking for a general abstract description of the site of test objects, before passing to the topos over it.

Instead of regarding affine varieties as formal duals of algebras over a 1-theory (the theory of ordinary commutative rings) you want to regard them as formal duals to algebras over a 2-theory –the theory of certain commutative 2-rings.

It may or may not be of interest to notice that this point of view on affines as formal duals to categorical rings has a long tradition and goes by the name of “categorical (nc-)geometry” or “derived noncommutative geometry”.

As far as I am aware the idea originates in the article

A. Bondal. Non-commutative deformations and Poisson brackets on projective spaces Preprint no. 67, Max-Planck-Institut, Bonn 1993, 1993.

It was further developed by Orlov, Keller and others and then turned by Kontsevich into a full-fledged program on categorical geometry (A A_\infty-categorical geometry, for him, they go all the way from 2-rings to (,2)(\infty,2)-rings with this idea).

The 2-ring version of the definition is for instance on page 24 here, which also points to the references by Bondal, Orlov, etc. that I mentioned.

One of the main applications of this “algebraic 2-geometry” is mirror symmetry: by Kontsevich’s conjecture (proven in some cases) two spaces that are mirror symmetry partners are actually equivalent when regarded as objects in this “algebraic 2-geometry”, namely their dual A A_\infty-categories of quasicohernt sheaves are equivalent.

All this seems relevant for the idea you are describing here, and I am glad we had this exchange now finally. I had tried this originally with a non-provocative question, but phrasing it provocatively has shown more reactions now.

Posted by: Urs Schreiber on December 10, 2010 3:07 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Urs wrote:

Instead of regarding affine varieties as formal duals of algebras over a 1-theory (the theory of ordinary commutative rings) you want to regard them as formal duals to algebras over a 2-theory – the theory of certain commutative 2-rings.

Maybe that’s true in that funny sense of ‘want’ where you can want something without knowing it. In mathematics it sometimes makes sense to say somebody wants something without knowing it. So maybe I want that (or wanted it back when I cared about this stuff) without knowing it.

But just to be clear: I never said anything about affine varieties; my remarks were about projective varieties. It’s projective varieties, not affine ones, that give rise to interesting theories in the doctrine of ‘dimensional theories’.

But affine varieties also give theories in this doctrine, of a rather degenerate sort — exactly as ungraded algebras are degenerate cases of graded algebras. So maybe that’s what you mean.

I am glad we had this exchange now finally.

I’m not, but at least it’s almost over.

Posted by: John Baez on December 11, 2010 11:10 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Good heavens. After attending a conference which dealt with some of the passionate debates of the nineteenth century mathematicians, I come back to the Café and find it happening here.

A feature of intellectual enquiry that intrigues me is when there are two frameworks dealing with the same topic (where ‘same’ may need some subtle treatment) and one claims superiority to the other. Do we find situations such as with chess, where if I play someone very good, they have complete understanding of what I’m doing? “You’re trying to do this, but it won’t work. You’d be better advised to do that, but that will just delay your losing.”

When teaching maths you can have that sense too. You understand the strategies to solve a problem, and nothing a student will do can surprise you as to these strategies. Jukka Keranen coined the term ‘cognitive control’ in his thesis – Cognitive control in mathematics, which captures the idea well.

It’s interesting to think whether absolutely greater cognitive control occurs when reading the mathematics of an earlier time. Probably not, but some wild claims have been made that large amounts of cognitive control have been lost. I was reassured by Matthew Emerton’s comment that we haven’t lost anywhere near as much control over nineteenth century geometry as the historian David Rowe claimed.

Proper control would have you be able to answer whatever problem could be answered, and explain what there is to explain, in the language of the other framework. You would also be able to point out in that same language what cannot be achieved without expansion of resources.

Even if all this were the case, it could well be worth developing the picture in the simpler framework for pedagogical reasons.

Posted by: David Corfield on December 13, 2010 11:05 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Ok, I certainly haven’t followed this in detail, but surely the point is that John and Urs are interested in slightly different things, and any “passion” is primarily arising from not initially spotting that?

Perhaps a better analogy than chess would be blackjack/21. You can develop game theory that talks about the information the various participants have to understand some things and generalise that to encompass other games like poker. You can also do the “inventing card counting” approach of finding a detailed set of procedures based on the rules and what you’ve observed that give you the best chance. This approach really doesn’t generalise to games beyond 21 effectively for various reasons, but that doesn’t mean that if you happen to playing blackjack the more general game theory is “superior”, it’s just looking at things with a different goal (namely not relying on the things unique to blackjack).

Posted by: dave tweed on December 13, 2010 2:45 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

but surely the point is that John and Urs are interested in slightly different things,

No, that’s not the point!

I am very much interested in what John and Jim are doing here. Otherwise I would not spend so much time with trying to find out what exactly it is and how exactly it relates to things that I and others already know. There is a reason why I did come back to this thread here, after searching for something related to my own research!

The 2-algebraic approach to geometry is of utmost importance. The point is that

  1. function 1-algebras do not have a general good pull-push theory in themselves, they need to be understood as decategorifications of something that does;

  2. function 2-algebras (such as categories of quasicoherent sheaves) do have a general good pull-push theory.

So it is of great importance to develop geometry based on formal duals of 2-algebras. This is, as I understand now, what Jim Dolan is doing, with focus on the strictly 1-categorical theory.

I dared to point out that Jim Dolan is not the first one to consider this kind of idea. Though certainly he will be the first one to do it his way. And I certainly believe that there is plenty of room for improving and systematizing accounts of this that do exist in the literature. If John’s writeup of Jim’s ideas produces such, I will be most delighted to see it.

At the same time I dare to have voiced a criticism: I said and still think that the intro to this work given here and on the wiki is somewhat misleading. This is now hardly controversial, is it: I was manifestly misled by it for some time. John seemed to have been annoyed by this, but that’s how it goes when one publishes a text: people will read the text and not the author’s mind.

Posted by: Urs Schreiber on December 13, 2010 3:05 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Ok, although just to be clear: just because someone is working with a different goal to oneself does not necessarily mean one wouldn’t be interested in what they’re doing. Indeed, I’m often interested in what people with different goals are doing in my area. The issues arise when one doesn’t immediately spot the different goals and automatically applies one’s own criteria.

My limited understanding from the thread was that you’re currently more interested in generalisation/unification whilst John is currently more interested in relating just algebraic geometry to category theory in order to understand/explicate the details of algebraic geometry. But that may very, very well be a wrong reading.

I was really responding to David’s implicit view (particularly with the chess analogy) that the most general theory is necessarily what everyone is seeking and will be used for all purposes. This is probably partly personal as someone who primarily works in finding useful specific structure that doesn’t generalise, but which is consequently more powerful than statements which are true in general.

Posted by: dave tweed on December 13, 2010 3:29 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Checking over what I wrote (as one does to check one hasn’t said anything inflammatory without realizing it), I see that I made an inopportune choice of words without realising it. The “interested in” in “John and Urs are interested in slightly different things” meant “John and Urs have objectives which are slightly different”. Then going on in a following comment to use “interested in” with both that meaning and the meaning “being happy to think about” in different paragraphs is very unclear writing. Hopefully it’s still decipherable what I meant.

Posted by: dave tweed on December 13, 2010 4:16 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Good heavens. After attending a conference which dealt with some of the passionate debates of the nineteenth century mathematicians, I come back to the Café and find it happening here.

A feature of intellectual enquiry that intrigues me is when there are two frameworks dealing with the same topic (where ‘same’ may need some subtle treatment) and one claims superiority to the other.

I can’t quite find my experience reflected in this. David, please do me a favor: summarize in your words what you think I said, what you think the “passionate debate” is.

I think I was just trying to understand what is advertized in this thread (which I think I do now), and making some remarks on what I think it is related to.

I am puzzled. Was my

what is it that is new here? I am asking this provocatively, but I am genuinely interested in hearing the answer

so offensive? Isn’t that an entirely normal question in reaction to the presentation of some work? It is the very first question every journal referee asks. It seems to me that every author should expect this question and should be happy to hear it: it gives occasion to point out again what the author thinks is good about his work.

I imagine that in a different universe, John or Jim could just have replied to that question. Here is a suggestion for what such a reply could look like:

What is new here is that we have a fresh systematic look at the description of varieties as formal duals not just to 1-algebras, but to 2-algebras: their monoidal linear categories of quasi-coherent sheaves. Instead of passing to the full (,2)(\infty,2)-algebras of complexes of quasicoherent sheaves that are discussed in the literature, we focus on the 1-category theoretic properties of these objects and instead of thinking of them as algebras over 2-theories, we find it useful to think of them as algebras over doctrines , which means almost the same but has some slightly different connotation. It seems to us that these 1-categorical aspects of categories of coherent sheaves have not received due attention so far. Specifically we notice that for projective varieties, these 2-algebras of quasicoherent sheaves are characterized by nice category-theoretic universal properties.

Posted by: Urs Schreiber on December 13, 2010 1:46 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Ok, ‘passionate debate’ wasn’t the right term. I was surprised by John’s overt expression of annoyance and then at your persistence in the face of that uncharacteristic expression.

It seems to me that quite a part of the trouble between people in maths comes from one party begrudging another party claiming cognitive control over their field, whether one believes the claim justified or not. In a world of perfected intellectual virtue, where truth is the only end, one should be happy to expose one’s theories to the scrutiny of other parties, and help in the process of discovering whether or not there is anything in one’s theories which provides unique insight. And one should be pleased to be shown the falsity of one’s belief in this unique insight, if indeed it is false.

Posted by: David Corfield on December 13, 2010 2:11 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

I imagine that in a different universe, John or Jim could just have replied to that question.

I thought Jim did?

Posted by: Todd Trimble on December 13, 2010 3:04 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

But just to be clear: I never said anything about affine varieties; my remarks were about projective varieties.

Okay, fine, so you are interested in projective varieties.

I clearly can’t fail to notice how annyoed you are with me, for the way in which I am being slow with understandig what the goals of this approach are, but at least I am genuinely trying to understand what is going on. That involves trying to figure out where you are just rephrasing known material and where you are suggesting something new. Sorry to hear that you find this annoying. There is also the possibility to regard it as a means to clarify things.

Looking back at this entry and the wiki page, I can’t help but feel that it is barely a surprise that I did not realize that you are not speaking about the topos-theoretic formulation of algebraic geometry and not about Grothendieck’s way of formulating algebraic geometry: this is after all what forms the leading paragraph of this entry. And on Jim Dolan’s wiki page it says

algebraic geometry and topos theory were for me two puzzle pieces that were supposed to fit but didn’t, two cultures that were supposed to communicate but didn’t. but now i have an idea for how they fit together, with the help of a missing puzzle piece called “the doctrine of dimensional theories”, and i want to try to explain it here.

I got away from all this with the impression that you are meaning to present a new way to understand the general abstract formulation of algebraic geometry in terms of topos theory. That, yes, made my eyebrows rise: the relation between topos theory and geometry (and in particular algebraic geometry) is well understood and there are good texts exposing the relationship.

Now after we talked about it, you emphasize that you are just interested in describing the classical thery of projective varieties in a category theoretic way. From what I understand, it is not quite actually the classical theory that you are describing. What Jim Dolan is actually doing, as far as I understand now, is having a close look at the “algebraic 2-geometry”-description of spaces: not as formal duals to 1-algebras of functions, but as formal duals to 2-algebras of coherent sheaves.

It took me a bit to realize this clearly, and with hindsight maybe I was being dense, but also I think the “doctrine”-language didn’t quite contribute to making a simple point clear. I can see why Jim likes that language. But it is also true that first, when I still thought this here was about topos-theoretic geometry, I thought the doctrines were supposed to play a role different from what I now understand they do: as I mentioned, in places such as Structured Spaces one conceives geometry generally in toposes over sites of formal duals of algebras over \infty-theories. The “doctrines” here are special 2-theories and you are interested in describing formal duals to some of their 2-algebras (say those corresponding to projective varieties for the moment).

I think that’s interesting. I wish we would discuss this. I don’t think that this has been discussed clearly before in this thread here and I’d be surprised if I am really the only reader of this thread who did not see this point clearly.

I also think it should be useful to compare to existing work based on the same ideas. There is a little “and now we do it right”-attitude in between the lines of your and Jim Dolan’s exposition. In a field such as algebraic geometry where many people have thought deeply about the foundations, such an attitude (which may well be justified) does well with going along with some comparison to what other people thought about before. If only it helps your readership to put your claims in perspective.

John, you and me we don’t have to talk about this. But maybe others here are interested in discussing this further. Or maybe not.

Posted by: Urs Schreiber on December 13, 2010 10:24 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

this is a very preliminary reply, directed more to the whole recent renewal of discussion in this thread than to the particular message that i’m attaching it to. it’s a sort of stopgap reply to try to prevent the discussion from getting too much out of hand before i can try to write some more informative and responsive reply.

i think that the discussion was threatening to get out of hand because of some accumulated misunderstandings. there’s a particular source of such misunderstanding that’s probably not visible to most of the people involved in the discussion, and perhaps i can counteract some of the misunderstanding by explaining a bit about this.

the basic misunderstanding concerns the nature of various collaborations between john baez and myself over the years. although these collaborations have had some interesting results, i’ve generally been dissatisfied with them because not much really resembling my actual views (usually quite different from john’s) has been communicated through them.

of the distorted impressions that people have received, the most serious is that i don’t have a viewpoint different from john’s, or else that i’m in some way reticent to present it. i have no such reticence; rather, it’s just that my personal style is to present my ideas in spoken rather than written form. i’m basically a classroom teacher, and i find writing, with its limited capacity for interaction and feedback, to be generally not a useful form of communication.

knowing of my dissatisfaction with previous collaborations, john agreed to try to collaborate with me once more, but with different ground rules that may have made him feel some obligation to defend my views as though they were his own. i can well understand how that might be a frustrating position to be in, since it’s similar to the position that i’ve found myself in in the past.

thus i suspect that some of the misunderstandings here arise from john being in this uncomfortable position. in any case, i advise against forcibly interrogating him to try to get him to defend my viewpoint, and against drawing conclusions from his answers about what my viewpoint actually is.

Posted by: james dolan on December 14, 2010 12:29 PM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Did UCR change their website? It looks like both the “streaming” and “downloadable” links are dead, thus I can’t access any of Dolan’s videos.

Posted by: Trent on December 24, 2013 9:46 AM | Permalink | Reply to this

Re: Algebraic Geometry for Category Theorists

Never mind… the links on the page that this post links to are dead, but I googled around and found an updated page with working video links: http://ncatlab.org/johnbaez/show/Doctrines+of+algebraic+geometry . Videos so small that the blackboard writing is almost entirely illegible… but hey, I’m still thrilled that videos on algebraic geometry for category theorists exist.

Posted by: Trent on December 24, 2013 9:57 AM | Permalink | Reply to this

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