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May 6, 2009

nLab - More General Discussion

Posted by David Corfield

With the previous thread on nLab reaching 343 comments, it’s probably time for a new one.

Let me begin discussions by asking whether it is settled that distributor be the term preferred over profunctor. I ask since it would be good to have an entry on the 2-category of small categories, profunctors and natural transformations. Should it be DistDist or ProfProf?

Posted at May 6, 2009 9:06 AM UTC

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Re: nLab - More General Discussion

I ask since it would be good to have an entry on the 2-category of small categories, profunctors and natural transformations. Should it be Dist or Prof?

I am inclined to follow the entry on distributor and say SetModSet Mod instead of DistDist or ProfProf.

I think, generally, if one can help it is useful not to name categories after their morphisms instead of after their objects.

So one should ask: which role do locally small categories play if I allow morphisms between them to be “distributors”=”profunctors”=”bimodules”?

The answer is: in that case they behave like (bases for) modules. So the 2-category they form should be called a 2-category of modules. Of SetSet-modules in the case of locally small categories. Of VV-modules in the general case.

By the way, there are entries on [[profunctor]] and [[bimodule]], too. At the moment [[profunctor]] just redirects, while [[bimodule]] mentions information pretty much overlapping with [[distributor]].

Posted by: Urs Schreiber on May 6, 2009 10:38 AM | Permalink | Reply to this

Re: nLab - More General Discussion

I second Urs’s suggestion. ModMod carries the basic intuition, and SetSet indicates the base of enrichment. ModMod seems to be favored in certain parts of the world where they talk about these things a lot (such as Sydney).

Between “profunctor” and “distributor”, I’ve more often seen “profunctor” and ProfProf, and the prefix “pro-” has lent itself to allied terms such as “promonoidal”. (I’m not too fussed about the fact that “pro-” is also used differently, as in “profinite”; this type of thing happens all the time.) I have to say that “distributor” doesn’t do much for me in terms of intuition.

Posted by: Todd Trimble on May 6, 2009 2:12 PM | Permalink | Reply to this

Re: nLab - More General Discussion

in re: I have to say that “distributor” doesn’t do much for me in terms of intuition.

cf. associater

Posted by: jim stasheff on May 6, 2009 2:38 PM | Permalink | Reply to this

Re: nLab - More General Discussion

“Associator” I like: as a noun back-formation from “associative”, it gives me just the right impression (of denoting a structure which witnesses associativity). But “distributor” in the sense of profunctor: I don’t see how it is similarly witness to distributivity.

Maybe the association should be instead with “distributions”, i.e., generalized functions. I’d have to think about that some more to see if it carried much weight with me.

Posted by: Todd Trimble on May 6, 2009 6:40 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Another sensible term for “profunctor” is correspondence. Used for instance pp 86 here.

Posted by: Urs Schreiber on May 6, 2009 8:09 PM | Permalink | Reply to this

‘profunctor’ vs ‘distributor’

I find that discussion about terminology helps me to clarify the ideas. Just as good terminology helps one to talk and think about things, so discussion about terminology forces me to clarify how to best talk and think about things.

As for the matter at hand, I have no preference between ‘profunctor’ and ‘distributor’, but I don't like ‘bimodule’ or ‘module’, for the reasons John gives (although SetModSet Mod seems fine, since that gives the proper context). In particular, our page bimodule really ought to be about the more general concept (like module is).

Posted by: Toby Bartels on May 7, 2009 7:52 PM | Permalink | Reply to this

Re: ‘profunctor’ vs ‘distributor’

I wrote:

In particular, our page bimodule really ought to be about the more general concept (like module is).

But my adjective ‘general’ is really not appropriate here, is it? They're both special cases of each other, like monoids and monoidal categories.

Posted by: Toby Bartels on May 8, 2009 7:57 PM | Permalink | Reply to this

Re: ‘profunctor’ vs ‘distributor’

They’re both special cases of each other, like monoids and monoidal categories.

Yes, as long as you interpret the one suitably generally. That is, a module for a monoid object in a monoidal category is a special case of a module for an enriched category where the category has one object, and likewise for a bimodule. However, a (bi)module for an enriched category is a special case of a module for a monoid object—as long as you allow your monoid object to live in a bicategory rather than merely a monoidal category.

Posted by: Mike Shulman on May 8, 2009 9:01 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Is there a reason why this isn’t taking place on the n-forum? The point of that was to make it easier to thread and so that people other than the “Big Three” could initiate subjects. It can even typeset simple LaTeX now.

Reminder of the url:

http://www.math.ntnu.no/~stacey/Vanilla/nForum

Posted by: Andrew Stacey on May 6, 2009 11:45 AM | Permalink | Reply to this

Re: nLab - More General Discussion

I suppose it’s a question of needs and visibility. Anyone can initiate a discussion from this post, and some from the last post, e.g., on discretism, were of a Café-like nature.

People go where they choose.

Posted by: David Corfield on May 6, 2009 1:58 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Actually, that makes my point. The n-forum is more visible from the n-lab than these discussions (at least, at the current revision of the home page). It is far easier to see what topics have been and are being discussed on a forum than in the threads of a blog. The forum has RSS feeds so it is just as easy to check as the cafe - indeed, a forum RSS feed is more informative than a blog one.

I agree that some threads in the last discussion were of a cafe-like nature and that’s why I haven’t posted “why not use the forum?” after each comment. The idea was that the forum was for non-mathematical discussion (though it does now have mathematical capabilities). I would still argue that the comments section of a thread on the cafe is not the best place for such a discussion once it reaches a decent length (whatever that may be). After all, it’s tricky to hunt it down if you weren’t aware of it from the beginning. But that’s a bit of a problem with the nature of the beast and probably not something to deal with here and now.

People go where they choose.

I’m amazed that you, being British (or at least living in Britain) can say this and keep a straight face! As the Xenophobe’s guide to the English puts it so eloquently: “The devotion of the English to queuing is such that an Englishman will join a queue without having a clue as to its purpose.”.

(lest I get into trouble yet again, let me point out that that is intended humourously. Even humorosly if you prefer.)

More seriously, people don’t go where they choose. People go where everyone else is going.

I think that a forum-based system for discussions related to the lab is much the best format. But that’s just my opinion and I’m nowhere near central in either the cafe or the lab clique. However, what is worst is having a mish-mash of lots of different places where discussion is going on with no clear demarkation. So if all discussion is to take place on these pages then the home page of the n-lab should be altered accordingly (and the forum taken down). If not, then the demarkation should be clear. What I had thought was:

  1. Discussion relevant to a particular entry takes place on that entry’s page.
  2. Discussion of a more general mathematical nature takes place in the threads of the cafe (in the unstated hope that a cafe patron will take up the topic and convert it into a major post).
  3. Discussion of a more technical/less mathematical nature takes place on the n-forum (such as what exactly does “heuristic” mean).
Posted by: Andrew Stacey on May 6, 2009 3:57 PM | Permalink | Reply to this

Re: nLab - More General Discussion

I think I agree with Andrew here.

I see the Café literally as a place where we meet and discuss things, and as the Lab as the place where we write up the results of discussion, or prepare notes to back up discussion or compose other texts of a nature different from a blog discussion.

And the n-forum as a place for meta-discussion about operating the Lab.

(On the other hand, my picture of Café-Lab interaction is not yet really approximated by reality.)

One minor comment:

(in the unstated hope that a cafe patron will take up the topic and convert it into a major post)

As far as I am concerned, I am generally happy to forward guest posts. But nobody should rely on me sensing unspoken wishes for guest posts. If you want one, please send me an email. Preferably one that contains the text in a form that I just need to copy-and-paste it.

Posted by: Urs Schreiber on May 6, 2009 4:38 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Ah, Andrew, one thing: it seems that the links to the RSS feeds on the nn-forum home page are broken.

Posted by: Urs Schreiber on May 6, 2009 4:52 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Take the title of this post then to be shorthand for ‘General discussion of issues arising from nLab of the kind of conceptual nature that are often discussed at the Café’.

On the other hand, it’s not always clear when what starts out as a fairly technical issue about a decision regarding nLab spins into a conceptual discussion. For example, discussions about terminology often have interesting associated baggage.

Posted by: David Corfield on May 6, 2009 5:28 PM | Permalink | Reply to this

Re: nLab - More General Discussion

“The devotion of the English to queuing is such that an Englishman will join a queue without having a clue as to its purpose.”

Alternatively, if needed for a purpose he comprehends, an Englishman will form (ie, commence) a queue, e.g. waiting for a bus.

Posted by: jim stasheff on May 7, 2009 12:44 AM | Permalink | Reply to this

Disproofs of Bell, GHZ, and Hardy Type Theorems and the Illusion of Entanglement

Hey guys, awesome site you have going here, its a real credit to you all: ) anyways, to go massively off topic from the subject of the thread (please excuse the impertinence but i was unsure of where else to put it lol) I was wondering what your opinions where on the work of Joy Christian (of Oxford and the Perimeter Institute) on entanglement, particularly his latest paper ‘Disproofs of Bell, GHZ, and Hardy Type Theorems and the Illusion of Entanglement’ ( http://arxiv.org/abs/0904.4259 ). Your thoughts and commentary on the matter would be greatly appreciated. Keep up the great work guys and stay safe : )

Posted by: Chris on May 6, 2009 3:58 PM | Permalink | Reply to this

Re: nLab - More General Discussion

The choice of profunctor vs. distributor is political insofar as ‘distributor’ is what French mathematicians use, while ‘profunctor’ is what the Australians use.

One advantage of ‘profunctor’ is that the prefix ‘pro-’ can be extended to other related contexts. For example, ‘promonoidal categories’ are very important (with examples coming from Day convolution).

So, between ‘distributor’ and ‘profunctor’ I prefer ‘profunctor’.

On the other hand, Ross Street and others have more recently taken to calling profunctors ‘bimodules’ or simply ‘modules’. I find these a bit vague, particularly the latter. I understand that a profunctor is a special case of a bimodule between monoid objects, and a bimodule is in turn a special case of a module. But, I don’t think it’s good to routinely denote a concept by the name of the most general concept of which it’s a special case! When I hear ‘module’ I think about all sorts of things; when I hear ‘profunctor’ or ‘distributor’ I instantly know what’s being referred to.

On the other hand, I don’t mind if people use SetModSetMod for the 2-category whose morphisms are profunctors. You might have to tell someone what this means, but after you do, they’re not likely to mix it up with anything else.

Posted by: John Baez on May 6, 2009 4:31 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Am I the only one who minds the clash of ‘profunctor’ with pro-object? If it were a completely different place in mathematics, I wouldn’t mind, but pro-objects in a functor category seem like something one might naturally consider. Moreover, since every presheaf is a colimit of representables, a distributor is really more like an ind-object in the functor category (though it need not be a directed colimit).

Re: correspondence, I would rather we not introduce a completely new word for something that already has three names—especially a word that is frequently used informally in mathematical discussion to mean something completely different.

Re: distributor, yes, ‘distribution’ is the intended intuition: just as a distribution is a generalized function, so a distributor is a generalized functor. I don’t know that this really gives me much intuition, but at least the word is unambiguous, unlike the other two candidates.

Re: naming categories, I think it is a good rule to name (nn-)categories after their objects (0-cells) when the morphisms (1-cells) in the category are ‘functions’ between these objects (as in SetSet, GrpGrp, CatCat, etc.), but when the 1-cells are ‘objects’ in their own right that happen to be ‘indexed’ by pairs of 0-cells (as in DistDist, RelRel, nCobn Cob, etc.) I think it is not inappropriate to name the category after the 1-cells.

Posted by: Mike Shulman on May 7, 2009 6:59 AM | Permalink | Reply to this

Re: nLab - More General Discussion

Re: distributor, yes, ‘distribution’ is the intended intuition: just as a distribution is a generalized function, so a distributor is a generalized functor.

I am happy with that, and this kind of preference of “distributor” over “profunctor” made me create the entry with title “distributor”.

On the other hand, the above analogy is a bit weak in that even though distributors generalize functors, the way they do so is not analogous to the way distributions generalized functions, is it?

And I always thought that the way distributions generalize functions is the reason for the word “distribution”, as in “probability distribution” where “events” are literally “distributed” over a probability space.

One last thing: I am not sure if I want to be opting for adopting the term “correspondence” officially, but among all the terms it might actually be the one which is best descriptive: the full sub-bicategory of SetModSetMod on discrete Set-categories is precisely the bicategory of sets and “correspondence spaces” in sets, aka spans.

Anyway, it’s all not so important. We should all save our energy for more substantive discussions than those about terminology…

Posted by: Urs Schreiber on May 7, 2009 9:56 AM | Permalink | Reply to this

Re: nLab - More General Discussion

Urs wrote:

We should all save our energy for more substantive discussions than those about terminology…

That’s sort of like saying we should spend less time watching sports so we can focus on more important news. Perhaps true, but not human nature. Mathematicians — and especially category theorists — find it fun to discuss terminology. I’m not sure why.

I’m very busy these days, so a discussion of terminology makes a nice relaxing break — unlike, say, a discussion of derived \infty-stacks.

Posted by: John Baez on May 7, 2009 2:41 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Nothing against relaxing once in a while.

It’s still frustrating that all the cool things that would be on-topic for a research blog find so little active attention here. It should be the opposite of an extra burden, as it seems to concern the research work of most of us in one way or other. I’d find it helpful instead of a burden to see derived \infty-stacks sorted out in discussion here.

What I’d find considerably relaxing in a time where I am busy with other things were a stronger feeling of a small but fine online community joined here behind me, which joins forces on research on, let’s see, what was the title of the blog?, ah, right nn-categories in math and physics.

I can’t meet Freed, Hopkins, Lurie and Teleman to chat with them, but I have a research blog that many interesting people participate in. If only we could get past the discussion of terminology, the potential synergy might actually make a difference. While we are sorting out terminology other collaboration units are solving the problems that we would have seemed to be the canonical community to look into together. But didn’t.

Posted by: Urs Schreiber on May 7, 2009 4:56 PM | Permalink | Reply to this

Re: nLab - More General Discussion

A span doesn’t look very much like a correspondence to me; if anything I would say a relation, or actually a bijection, is more like the way I use “correspondence” informally.

You’re right, though, that the relation between distributors and functors is not much like the relation between distributions and functions in any way other than “is a generalization of.”

Regarding terminology, I actually think using good terminology is really important. Among other things, category theory is a language for mathematics, and so, among other things, category theorists are linguists. So it’s not surprising, nor, I think, to be denigrated, that we spend time thinking about the best names for things.

Posted by: Mike Shulman on May 7, 2009 7:24 PM | Permalink | Reply to this

Re: nLab - More General Discussion

A span doesn’t look very much like a correspondence to me

Many people in geometry will know “span” only as “correspondence space”. There it is the standard term.

Notably for instance in the context of Fourier-Mukai transformations and, by inductions, in geometric \infty-function theory! #

When I say “span” around here, people will ask me “Ah, you mean correspondence space?”

I actually think using good terminology is really important. Among other things, category theory is a language for mathematics, and so, among other things, category theorists are linguists. So it’s not surprising, nor, I think, to be denigrated, that we spend time thinking about the best names for things.

Yes, agreed.

Posted by: Urs Schreiber on May 7, 2009 8:14 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Ok, well, you learn something new every day. Who came up with that name?

Posted by: Mike Shulman on May 8, 2009 8:06 AM | Permalink | Reply to this

Re: nLab - More General Discussion

I do not know about “correspondence space” but the correspondences themselves are old and dominant terminology in geometry. Usual correspondences (set theoretically) are just a variant of the notion of a relation. Hence we have two projections from X x Y to X and Y forming visually a “span”. Then it generalizes to have general Z instead of X x Y.

Grothendieck talked in 1950-s about correspondences when started dreaming on motives in algebraic geometry, cf. pure motive (nlab). Nowdays the terminology is dominant in all main flavours of geometry, as opposed to categorical preference for spans.

To name few examples, Chow correspondences are built using formal linearization of suitably graded classes of cycles on X x Y (hence they are not spans in the original category of say varieties, but an additive and homological variant of the notion). Connes and Marcolli nowdays talk correspondences in nc geometry; it is building on a much earlier work of Connes and Skandalis explaining some bimodule issues in KK-theory. Even right today some papers appeared on the arXiv with Lagrangean correspondences in the title. In some areas people talk about “roofs” assuming drawing source Z of the two legs on the top; it is my impression than talking spans is mainly 1960-s habit from pure category theory and graph theory, maybe somebody knows earlier usages in other areas ?

Now one elementary but important question which I do not know a sensible answer to.
When one replaces schemes by categories of quasicoherent sheaves (including dreamed versions in nc geometry) one replaces as we all know morphisms by adjoint pairs of functors (geometric morphisms). However the reconstruction of spectra (form categories) is NOT functorial for such generalized/Morita morphisms, nor for maps induced by geneuine morphisms of nc rings, while it is at the level of center (thus for commutative cases like points of the commutative schemes and topoi it is OK). Hence although we can reconstruct schemes from spectra, and even commutative morphisms, with nc case and with anafunctors we have nothing. But anafunctors suggest, higher categorical picture. On the other hand, usual thinking of schemes, algebraic spaces etc. is as presheaves of sets on the category of affine schemes which are sheaves in some Grothendieck topology; moreover these presheaves are locally representable. Now if I replace schemes by categories, I still have reconstruction locally so I think of morally having locally representable situation. But with anafunctors as morphisms I can not make gluing of these representables; we just glue the abelian categories of sheaves, not sheaves of sets. However I believe that if one could somehow build simplicial presheaves in that case. So for nc case, gluing of quasicoherent sheaves by means of localization should give (by sort of reconstruction) at the level of anafunctors, infinity-categorical gluing of presheaves of simplicial sets, which are locally just representable presheaves of usual sets on NAff (that is opposite to the category of nc rings). This would give more complete meaning to the reconstruction problem as well as connecting the two approaches to nc spaces, where in one class of examples one has objects which are glued from pieces represented by categories of qcoh sheaves, while in another class one represents the spaces by some naturally defined functors. Moreover, I have a very promising example where the gluing of categories of sheaves is not flat, hence one has even at the input level derived version of descent, so there is no hope to represent the thing by set valued functor; the example is very natural as it explains geometrically some calculations of Gel’fand/Retakh on quasideterminants. It is sort of homotopy nc scheme I would say, so I would be very happy if the derived gluing of categories of abelian qcoh sheaves would translate to a higher descent of presheaves of simplicial sets on NAff.

Posted by: Zoran Skoda on May 12, 2009 7:38 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Zoran said: Hence although we can reconstruct schemes from spectra

oops I wanted to say Hence although we can reconstruct schemes and spectra from categories of quasicoherent sheaves/modules

Posted by: Zoran Skoda on May 12, 2009 7:51 PM | Permalink | Reply to this

profunctors vs correspondences

Having just spent over a week listening to people at the TFT conference talk about “correspondences,” I still think that it makes less sense than “span” but I accept its usage as a fait accompli. (-:

However, at the moment I actually think that this other use of “correspondence” argues against reusing it to mean profunctor/distributor/module, because they are not quite the same, but close enough in meaning to be confusing. In particular, profunctors in SetSet can be identified with two-sided discrete fibrations, which are a particular sort of span in CatCat. But more general spans in CatCat (and especially in GpdGpd) are also important in some contexts. So if we say “correspondence” to mean “profunctor,” then people for whom “correspondence” means “span” might think we mean a span in CatCat, and it might take them a while to realize their mistake. (Whereas if we say “profunctor” or “distributor” they will at least be prompted to ask “what’s that?”)

Perhaps also of note is that lots of people at the TFT conference are using “module” to mean (quite sensibly) one category equipped with an action of another monoidal category. Which is, of course, a completely different notion from a profunctor.

(As you may be able to guess, I am gradually becoming a bit more favorably inclined towards “profunctor,” perhaps partly because I currently have two coauthors who use it. The general applicability of “pro-” is, admittedly, a nice feature, and “distributor” is, admittedly, a fairly ugly word.)

Posted by: Mike Shulman on May 26, 2009 3:00 AM | Permalink | Reply to this

Re: profunctors vs correspondences

Having just spent over a week listening to people at the TFT conference

So how was it??

at the moment I actually think that this other use of “correspondence” argues against reusing it to mean profunctor/distributor/module, because they are not quite the same

As the recent discussion on the nn-Cat mailing list reminded us: might it be best if we think of profunctors entirely in terms of concontinuous functors on presheaf categories, and then further as cocontinuous functors on [[presentable (,1)(\infty,1)-categories]]?

That should also make the relation to correspondences of the kind I am guessing they talked about at Northwestern transparent and clear, using Ben-Zvi/Francis/Nadler’s theorem that all (“perfect”) pull-push correspondences are given by such cocontinuous (,1)(\infty,1)-functors (as around the highlighted box in section 4 of [[geometric \infty-function theory]]).

Posted by: Urs Schreiber on May 26, 2009 9:48 AM | Permalink | Reply to this

Re: profunctors vs correspondences

Having just spent over a week listening to people at the TFT conference

So how was it??

Pretty good, at least the workshop week. I got lost pretty quickly in the actual conference, but it made me want to learn more about all this stuff.

might it be best if we think of profunctors entirely in terms of concontinuous functors on presheaf categories

My answer would be “no,” at least “no” specifically to your use of the word “entirely.” That is certainly one useful way to think of profunctors, but I find that thinking of them as bimodules is frequently much more useful.

Posted by: Mike Shulman on May 27, 2009 11:18 PM | Permalink | Reply to this

enriched

While we are talking # about enriched category theory, I have a related question. I am looking for the goood way to think of the following kind of situation:

Suppose VV a closed symmetric monoidal category with coproducts that are respected by the tensor product, (ab)c(ac)(bc)(a \coprod b) \otimes c \simeq (a \otimes c) \coprod (b \otimes c).

Let BV\mathbf{B} V be the corresponding bicategory and consider for any ordinary category CC a lax functor

F:CBV. F : C \to \mathbf{B} V \,.

This is something close to a VV-enriched category: for each (afb)Mor(C)(a \stackrel{f}{\to} b) \in Mor(C) there is F(a,f,b)Ob(V)F(a,f,b) \in Ob(V) and composition operations

F(a,f,b)F(b,g,c)F(a,gf,c) F(a,f,b)\otimes F(b,g,c) \to F(a,g\circ f, c)

etc.

Okay, now the observation that I am looking for comments on (nothing profound, but anyway):

Let 1:CBV1 : C \to \mathbf{B} V be the constant functor.

Then lax transformations

η:F1 \eta : F \Rightarrow 1

consist of a collection of morphisms in VV

η f:F(a,f,b)η bη a \eta_f : F(a,f,b) \otimes \eta_b \to \eta_a

satisfying some condition. The hom-adjunct of these (I am assuming VV to be closed monoidal, recall) is

η¯ f:F(a,f,b)[η b,η a] \bar \eta_f : F(a,f,b) \to [\eta_b,\eta_a]

and the conditions say that this can be read an an “enriched functor”

η¯:Graph(F)V, \bar \eta: Graph(F) \to V \,,

(When CC is a codiscrete category this is literally an enriched functor of VV-enriched categories.)

So η¯\bar \eta is a module over Graph(F)Graph(F).

When this arises in practice, and when VV has a 0-object, people like to rephrase this by forming the objects in VV

A:= (afb)Mor(C)F(a,f,b) A := \coprod_{(a \stackrel{f}{\to}b) \in Mor(C)} F(a,f,b)

and

K:= aObj(V)η a K := \coprod_{a \in Obj(V)} \eta_a

and consider AA as an algebra internal to VV with product in components

F(a,f,b)F(c,g,d){F(a,gf,c) ifb=c 0 otherwise F(a,f,b)\otimes F(c,g,d) \to \left \lbrace \array{ F(a,g\circ f,c) & if b = c \\ 0 & otherwise } \right.

the composition if defined and 0 otherwise. Similarly KK becomes a module over AA

η¯:AKK. \bar \eta : A \otimes K \to K \,.

The famous example that I am thinking of are linear groupoid representations as they are considered in the context of Drinfeld doubles.

In that case CC is some finite groupoid, V=VectV = Vect, F:CBVectF : C \to \mathbf{B}Vect sends all morphisms to the tensor unit and on 2-cells is a groupoid 2-cocycle α\alpha.

Then a transformation η:F1\eta : F \to 1 is an α\alpha-twisted linear representation of CC.

But instead of saying it this way, people like to form the α\alpha-twisted groupoid algebra of CC and say that η\eta is a module over that. This being a special case of the general construction I just tried to describe.

I am thinking there should be some standard enriched-category theoretic way to think of this passage from lax transformations to modules over algebras. Probably I shouldn’t post this here but think about this a bit more myself. But anyway.

Posted by: Urs Schreiber on May 6, 2009 9:13 PM | Permalink | Reply to this

Isbell envelope and structured space

In the context of our discussion of smootheology, may I draw your all attention to the following aspect, maybe:

there is some conceptual similarity between the kind of considerations that Andrew Stacey wrote about at [[Isbell envelope]] and the basic idea of [[structured generalized space]].

Notice, as highelighted at that nnLab entry, that when you strip the idea of “structured generalized space” off all of its technical baggage, the fundamental concept here is that of a bi-presheaf or whatever you call it, anyway a (\infty-) functor

X:S op×CV, X : S^{op} \times C \to V \,,

where VV is our enriching category (Set or SimpSet, notably) which we are to think of as the functor which describes a generalized space XX by telling us for each USU \in S and RCR \in C the collection of morphisms from UU to RR that factor through XX.

One needs to not let oneself be distracted by the fact that Jacob Lurie and David Spivak in their texts here tend to take SS to be open subsets of a given ordinary space XX' of sorts. The same logic works more generally, and that case only corresponds to generalized spaces “over XX'”.

On that rough level it seems to me that the [[Isbell envelope]] idea would provide a kind of refinement of such a “structured space modeled on a gros topos”, in that for the case S=CS = C (probes being the same as co-probes) probably one would want to identify the XX above with the [[image]] of the natural transformation

c:P×FS(,) c : P \times F \to S(-,-)

that is part of the definition of [[Isbell envelope]]

because that’s precisely the formal statement corresponding to the idea that “PP gives the collection of maps from SS into the generalized space, FF that out of it, and XX the maps factoring through it”.

I suggest with that picture in mind go and re-read [[structured space]] and see what happens.

Posted by: Urs Schreiber on May 12, 2009 9:22 PM | Permalink | Reply to this

Re: Isbell envelope and structured space

Above I tried to draw the inclined blog reader’s attention to the observation that there is close similarity between the “Frölicher presheaves” we were discussing here in general and Andrew Stacey’s formalization in terms of [[Isbell ebvelope]] in particular, and Jacob Lurie’s definition of [[structured generalized space]].

I think there is even more to it. I am now going to present a little observation on how this relates to differential cohomology, and to codescent and the van Kampen theorem.

It’s just a trivial observation on how concepts match, but might be useful.

So, recall, for instance from the brief discussion at [[differential nonabelian cohomology]], that with \infty-stacks on the site DiffDiff – i.e. with a notion of smooth \infty-groupoid – given, we can start talking about differential cohomology as soon as we fix a functor

Π:DiffStacks(Diff) \Pi : Diff \to \infty Stacks(Diff)

which we think of as the functor that sends each manifold to its smooth [[fundamental \infty-groupoid]].

So for instance when we model \infty-stacks by simplicial presheaves we would tend to define for each UDiffU \in Diff the \infty-groupoid Π(U)\Pi(U) to be the \infty-stack given by

Π(U):VDiff(V×Δ ,U). \Pi(U) : V \mapsto Diff(V \times \Delta^{\bullet}, U) \,.

But other choices are possible, too.

The main point is that we want Π\Pi to satisfy some kind of homotopy van Kampen theorem, which roughly says that for {Y αU}\{Y_\alpha \to U\} a covering family, we are able to glue the Π(Y α)\Pi(Y_\alpha) to something that is equivalent to Π(U)\Pi(U).

With that in mind, return to Lurie’s def 1.2.8 in Structured Spaces to find that we may essentially say that

Π\Pi is a DiffDiff-structure

or in other words

Π\Pi defines an \infty-structure sheaf on DiffDiff.

Given any \infty-topos HH with structure sheaf OO in this sense, we can say that the OO-[[cohomology]] of an object XX in HH is the cohomology of O(X)O(X)

H O(X,A):=H(O(X),A). H_O(X,A) := H(O(X),A) \,.

If we do this for O=ΠO = \Pi we get the notion of flat differential nonabelian cohomology. Given any \infty-topos HH with structure sheaf OO in this sense, we can say that the OO-[[cohomology]] of an object XX in HH is the cohomology of O(X)O(X)

H O(X,A):=H(O(X),A). H_O(X,A) := H(O(X),A) \,.

If we do this for O=ΠO = \Pi we get the notion of flat differential nonabelian cohomology.

Now, that said, there is one extra property of Π\Pi that I can’t as yet quite recognize in the context of “structured spaces”.

Crucially, for each smooth \infty-stack there is a morphism

XΠ(X) X \hookrightarrow \Pi(X)

naturally in XX, which embeds XX as the constant paths in Π(X)\Pi(X). this is crucial for defining non-flat differential cohomology as the obstruction theory to lifting through the corresponding forgetful morphism

H(Π(X),A)H(X,A) H(\Pi(X),A) \to H(X,A)

that forgets the flat connection.

I’d find it nice to understand if that, too, may be understood as just a special case of a much more general theory of cohomology of structured generalized spaces. But maybe it’s better thought of as something particular to Π\Pi.

Posted by: Urs Schreiber on May 20, 2009 9:02 AM | Permalink | Reply to this

Re: Isbell envelope and structured space

This is one of those things I wish I could understand…

Posted by: Eric on May 20, 2009 6:42 PM | Permalink | Reply to this

Re: Isbell envelope and structured space

This is one of those things I wish I could understand…

This is one of those things that I just skim through, having the beginning and trusting that Urs did it right.

Eventually it goes to nLab, and then after I've had three or four chances to go through various versions and make minor grammatical and markup corrections, then I start to understand. (So the nLab is very useful, you see!)

Posted by: Toby Bartels on May 21, 2009 3:03 AM | Permalink | Reply to this

Re: Isbell envelope and structured space

I wrote

I’d find it nice to understand if that, too, may be understood as just a special case of a much more general theory of cohomology of structured generalized spaces

One can say at least this:

There is the trivial differential structure

Π 0(,)=Diff(,) \Pi_0(-,-) = Diff(-,-)

and to every other differential structure there is a local morphism

Π 0Π \Pi_0 \to \Pi

in the sense of definition 1.2.8 in “Structured spaces”.

In components this is the morphism XΠ(X)X \to \Pi(X). That it is local says pretty much, when you unwrap it, that XX is the inftyinfty-stack of constant paths, indeed.

Posted by: Urs Schreiber on May 22, 2009 6:23 AM | Permalink | Reply to this

Re: Isbell envelope and structured space

The heretic speaks:

will someone remind me why we care about smootheology?

to get the `right kind of’ differential forms?

and why do we so often restrict to manifolds?

even Poincare’ duality can hold more generally

Posted by: jim stasheff on May 23, 2009 6:35 PM | Permalink | Reply to this

Structured generlized space and chiral deRham complex

I am wondering:

the chiral deRham complex is a presheaf of vertex operator algebras.

Each such vertex operator algebra is thought to be equivalent to a local net of algebras. This is a co-presheaf of algebras.

So it might seem the chiral deRham complex wants to look like a co-presheaf valued presheaf.

Where the contravariant argument varies in target space, while the covariant one varies in parameter space.

Does this perspective ring a bell with anyone?

Posted by: Urs Schreiber on May 29, 2009 9:59 AM | Permalink | Reply to this

Re: Isbell envelope and structured space

I’m starting to work my way through all this. Not all the way there yet so these are comments “from the journey” as it were.

One good thing about giving talks on stuff is that it forces you to think about what the key idea is. Of course, each time you do this, you gain new insights and the old ones seem ridiculous. But I thought I may as well share this amazing insight with you.

Firstly, the trichotomy between “maps in”, “maps out”, “both” is a red herring. In the latest version of the paper, I showed that both “maps in” and “maps out” can be rewritten as “both”.

Secondly, and more importantly, “being Frölicher” is not really about saturation, it’s about morphisms. Everything can be put in terms of compatible bi-presheaves T i×T oST_i \times T_o \to S (where we allow different input and output test categories, and SS may not necessarily be SetSet if we want some other context). The key is the morphisms.

There are two choices for morphisms. One choice is that a morphism XYX \to Y is a pair of natural transformations C XC YC_X \to C_Y and F YF XF_Y \to F_X with suitable compatibility between them (when translating a “maps in” or “maps out” category into the “both” situation, one of these natural transformations completely determines the other). The other choice is that a morphism XYX \to Y is a composition natural transformation C X×F YHomC_X \times F_Y \to Hom, where Hom:T i×T oSHom : T_i \times T_o \to S is the “obvious” hom-functor. If you take this view then you are forced into a sort-of saturation condition in order to build a category.

This is my “amazing insight” from preparing the Ottawa talk. It might make it clearer what is the real choice between Frölicher spaces and the other ones, and how to generalise it to other contexts.

Posted by: Andrew Stacey on June 1, 2009 8:32 PM | Permalink | Reply to this

Re: nLab - More General Discussion

I disagree with moving nlab entries with ascii title, or title with less relevant accents, into the title with various straneg symbols. This makes less favorable indexing. Namely google puts pages with search item in the title naturally higher than concurring items; furthermore it is easier to write the url from the guess.
Tonight central European time, the server takes long times to respond, even over ten minutes for submitions of changes of items. What is going on ? I postponed my visit to family for half a day in order to work on nlab and everything gets stalled.

Posted by: Zoran Skoda on May 22, 2009 7:37 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Hi Zoran,

I’m suffering the same problem with delays.

I experimented with a few pages, i.e. moving them to the symbolic title version, but will change things back since you prefer the original way.

Part of the delay could be because several of us were editing various pages at the same time. I will take a break so that hopefully your experience gets better.

What that means is that the edits I made will remain for a while, but rest assured I’ll revert things back to the way they were later.

Posted by: Eric on May 22, 2009 7:50 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Eric continued the discussion here.

As you noted, I may still change some of the link displays, but without changing the links themselves, e.g. now and then, I may change a infinity-category to ∞-category so that it looks better in the page without changing the structure.

That should be fine. I tend to write \infty-category (and even 22-category) to get the math in math mode (which is distinguished by its font), but I don't have any strong opinion about that. (But note that nn-category is more noticeably different from n-category.)

Posted by: Toby Bartels on May 23, 2009 1:06 AM | Permalink | Reply to this

ASCII page titles

Note to all:

Discussion of this issue began here but really should continue here. (In particular, I did it wrong the first time.)

Posted by: Toby Bartels on May 22, 2009 8:54 PM | Permalink | Reply to this

Re: ASCII page titles

Now that we have working redirects, I’d like to revisit this issue.

I would like to redirect [[infinity-category]] to [[\infty-category]] and all such ASCII page titles to the corresponding unicode pages.

Currently, Toby has implemented a few redirects in the opposite direction. For example, currently [[\infty-category]] redirects to [[infinity-category]]. I’d prefer to end up on the nicer looking page.

Any objections?

Posted by: Eric on June 4, 2009 7:35 PM | Permalink | Reply to this

Re: ASCII page titles

Let me say that another way…

Now that we have redirects, it doesn’t matter much whether the content resides on [[infinity-category]] or [[\infty-category]]. For the sake of my eyes, I prefer the latter, but its not a big deal.

To make a change would require effort, e.g. renaming pages, inserting redirects, etc. I’m willing to extend the effort, but won’t do anything until there is a quorum.

For example, which page do we want to land on:

[[(infinity,1)-category of (infinity,1)-categories]]

or

[[(∞,1)-category of (∞,1)-categories]]?

Posted by: Eric on June 5, 2009 12:08 AM | Permalink | Reply to this

Re: ASCII page titles

Besides people's objections, Eric, another reason to hold off is that what you want will be different if your proposal is adopted to prioritise redirection commands over extant pages.

Posted by: Toby Bartels on June 5, 2009 12:39 AM | Permalink | Reply to this

Re: ASCII page titles

Any objections?

I suppose that Andrew will complain that the discussion should be on the Forum, but as I'm indifferent between there and this blog entry …

I don't think that you should go too crazy with the redirects yet, since the feature is not stable; things might change, and there may be bugs. You'll want whatever you do to be reversible.

Already I messed things up a bit with (∞,1)-category; I moved it to (∞,1)-category/history (and also edited the redirect commands there and at (infinity,1)-category to get things set up properly), but somehow I managed to create a new page (∞,1)-category as well. (The Lab went down in the middle of this, which I believe has something to do with that.) So I ended up duplicating a page.

I've only tried moving around pages that don't actually have any unique content, even in their edit histories. (Normally you can identify these as being in category: delete, although for some reason your symbol-named pages, Eric, are in the other category.) This is because deleting a page is far more serious than duplicating, but in fact it shouldn't be possible to do that accidentally. By the same token, however, duplicating is not reversible, so try not to do that either. (In particular, if the Lab goes down, don't just continue where you left off when it comes back up, like I did!)

The bottom line is that you can set it up so that links to [[infinity-foo]] go seamlessly to [[∞-foo]], so I have no objections to moving the former to the latter if you so set it up. I reserve the right to later argue that we ought to move it back, but it won't cause any harm now.

Posted by: Toby Bartels on June 5, 2009 12:25 AM | Permalink | Reply to this

Re: ASCII page titles

Toby proved prophetic:

I suppose that Andrew will complain that the discussion should be on the Forum, but as I’m indifferent between there and this blog entry …

Yup. I only came across this by accident. One more comment on the cafe and I wouldn’t have done so as it would have dropped off the list of recent comments.

Posted by: Andrew Stacey on June 8, 2009 8:26 AM | Permalink | Reply to this

Re: ASCII page titles

Any objections?

In any case, I object to fixing all of the current redirects wholesale until something is done to catch external links.

We would even break links on this very page! (The first example is the first link in the comment section.)

Posted by: Toby Bartels on June 5, 2009 3:23 AM | Permalink | Reply to this

Re: ASCII page titles

Over at the forum, I have suggested that we ask Jacques to implement an ability for a page to specify how its title should be displayed. For instance, there could be a [[!title ]] command at the top. This way we could have the content at infinity-category with a command [[!title $\infty$-category]] to make it display prettily. Assuming this is possible, this way even people who can’t type unicode could create nice-looking page titles.

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

Re: ASCII page titles

Over at the forum […]

That discussion is here.

Posted by: Toby Bartels on June 9, 2009 3:06 AM | Permalink | Reply to this

nLab server responsiveness

the server takes long times to respond

Yes, I know this keeps happening and have tried to improve on it, notably with the help of Andrew Stacey, but without much success so far.

It seems that apart from the memory leak that the “Mongrel” bit of the instiki software apparently still has, worse is that the server on which the nnLab is hosted is decidedly a bit buggy and apparently responsible for much of the low responsiveness we experience.

The weirdest thing that happened was a few days ago when Andrew Stacey and myself were simultaneously logged into the machine while at the same time communicating by email in order to figure out why the system is so slow. At some point the server started mixing up our terminal output: I saw Andrew’s keystrokes appear in my window – veery slowly – and Andrew similarly saw mine.

So , whatever that means in detail, it means we should look for another hosting server for the nnLab, eventually. Andrew said there might be a chance that he could host the system on his own equipment eventually.

Myself, I am neiter versed nor equipped sufficiently to easily take care of this quickly myself.

Posted by: Urs Schreiber on May 24, 2009 1:51 PM | Permalink | Reply to this

special symbols in entry names

Whatever we do about the special symbols in entry names itself, it would be desireable to have them in the text. For instance all the links in the text reading [[(infinity,1)-anything]] at the moment are getting on my nerves (even though I am responsible for most of them).

It should appear as [[(,1)(\infty,1)-anything]], whether by having a redirect entry or with by-hand redirect as [[(infinity,1)-anything|(,1)(\infty,1)-anything]].

Toby writes:

Can everybody figure out how to type ‘\infty’ without using iTeX or &-entities, which don’t work in links, and also without doing anything inconvenient like looking up Unicode or copy and paste?)

No, not everybody can. I can’t! :-)

We need a description at the [[HowTo]] page.

Posted by: Urs Schreiber on May 24, 2009 2:07 PM | Permalink | Reply to this

Re: special symbols in entry names

Another way to think of this is that if someone is too troubled to copy & paste a symbol, then they could actually reduce trouble overall by not pasting a link at all and leave linking to someone else.

If someone has content they would like to submit, I can’t imagine how the format of a link would be a deterrent when they have the option of not including links at all. I’m sure there are others out there, like me, who are happy to supply links when they think a link would be appropriate.

I guess I have a low tolerance pain and seeing “infinity-anything” even in a page title hurts my eyes and knowing an alternative is available makes me inclined to want to do something about it.

In the scope of world problems, this ranks pretty low, so the status quo is fine with me if it is fine with everyone else.

Posted by: Eric on May 24, 2009 4:29 PM | Permalink | Reply to this

Re: special symbols in entry names

Would this be useful on a HowTo?

  1. Install Linux (or BSD, Unix, etc).
  2. Install SCIM.
  3. Select Other -> Latex.
Posted by: Toby Bartels on May 24, 2009 8:07 PM | Permalink | Reply to this

Re: special symbols in entry names

I hope I am not more dense than average, but I need more instructions than that. Once SCIM is installed (it is apparently installed by default in recent versions of Ubuntu), how do I use it? I am failing to find documentation.

Posted by: Mike Shulman on May 25, 2009 4:33 AM | Permalink | Reply to this

Re: special symbols in entry names

I hope I am not more dense than average, but I need more instructions than that.

My instructions were brief because the post was a bit facetious; as you say in another comment, ‘I don’t think that instructions requiring you to install linux on your computer are “simple.” (-:’. And yet SCIM is not available for Windows or Mac OS 9 (although one might hope that there be an equivalent for Windows). I should have noted in my post that Mac OS X is a version of BSD, so it should work there.

Once SCIM is installed (it is apparently installed by default in recent versions of Ubuntu), how do I use it?

If your system's like mine, try System -> Preferences -> SCIM Input Method Setup. You’ll probably also have to install a whole bunch of Input Method Engines like I did; search for scim in the Synaptic Package Manager.

Of course, there's always the Character Map; all modern user-friendly operating systems have one of those, although it's not exactly the easiest thing to use.

Posted by: Toby Bartels on May 25, 2009 11:30 PM | Permalink | Reply to this

Re: special symbols in entry names

I’m still hoping someone will tell me how to use (not just configure) SCIM once I have it installed. I am failing to find any documentation.

Posted by: Mike Shulman on June 5, 2009 5:04 PM | Permalink | Reply to this

Re: special symbols in entry names

Eric wrote:

Another way to think of this is that if someone is too troubled to copy & paste a symbol, then they could actually reduce trouble overall by not pasting a link at all and leave linking to someone else.

Yes, but what if they want to create that page? They may be too timid to create the page at a ‘wrong’ name; in any case, somebody will have to move the page later. (I know that you're willing to do this, but are you also willing to fix all of the links to what is now a redirect page?) The naming conventions are designed to avoid this.

Urs wrote:

It should appear as [[(,1)(\infty,1)-anything]], whether by having a redirect entry or with by-hand redirect as [[(infinity,1)-anything|(,1)(\infty,1)-anything]].

This is impossible (without significant work by Jacques), but

[[(infinity,1)-anything|(∞,1)-anything]]

is possible. (The difference is subtle; check the fonts.) I've been doing

(,1)(\infty,1)-[[(infinity,1)-anything|anything]]

as a matter of course, although I don't fix everything that I see (especially in the many pages of your material!).

Posted by: Toby Bartels on May 24, 2009 11:54 PM | Permalink | Reply to this

Re: special symbols in entry names

I think maybe we are worrying too much about unlikely hypothetical situations. We have a simple way to do things the right (aesthetically pleasing) way. A few bullet points on a HowTo page could easily instruct any future author how to create (and link to) pages with symbols in the title.

Just being a devil’s advocate…

PS: Here are some pages

\infty-groupoid

\infty-category

(\infty,0)-category

(\infty,1)-category

ω\omega-category

strict ω\omega-categoy

strict ω\omega-groupoid

\infty-stack

Personally, I think these look a lot better and are worth any minor sacrifices.

Posted by: Eric on May 25, 2009 1:45 AM | Permalink | Reply to this

Re: special symbols in entry names

I don’t think the hypothetical situation is at all unlikely. And I don’t think that instructions requiring you to install linux on your computer are “simple.” (-: But even just asking you to install something like SCIM, just to be able to create links to important commonly-linked-to pages, is I think asking too much. (By the same token, I would prefer we avoid writing anything that looks like gibberish to people who don’t have special fonts installed.)

BTW, does anyone have instructions for a Windows or Mac user? I don’t need them myself, but their difficulty level is relevant to the discussion.

Posted by: Mike Shulman on May 25, 2009 4:07 AM | Permalink | Reply to this

Re: special symbols in entry names

There has been some discussion of this in latest changes, which I repeat here:

  • May 22, Zoran Škoda: By the way, the server is very erratic tonight, and having sometimes responses delayed by 5-10 minutes. On the other hand I strongly disagree with the changes of the names of entries massively being done today by Eric: he moves infinity-category into ∞-category. Though this is graphically appealing, google and others put higher in search results items which have search name in the title, and the entries they index are index basically by the ascii. I want to see nlab entries high in the google search, this makes our effort more useful. Fancy graphics can be WITHIN the entry, and prefereably in this decade still not in the title. I would do the redirects in the symbol variant of the title instead!! What the others think ?

  • May 22, Toby Bartels: I don’t think that we should change what works for us to fit Google. It’s Google's job to give good search results, not ours to optimise Google's finding us, which Google tries to prevent anyway. That said, I agree with you about the page names, for other reasons.

  • May 25, Zoran Škoda: as far as google (and other search machine’s) counterarguments of Toby I still disagree. I do care if our stuff is well indexed and hence better used by anybody looking for answer at the google including us; and I do not care about vanity issues of pro or contra google movements. If somebody gets directed to a less relevant page this is creating noise, and showing a less convenient side of our work. If effectivity is not important why are we doing this ?

Posted by: Toby Bartels on May 25, 2009 11:55 PM | Permalink | Reply to this

Re: special symbols in entry names

Although I agree with Zoran (at least for now, given the software that we have) about page names, I think that it's very dangerous to rely on arguments about search engines.

I am not on the Lab for the vanity of seeing my work ranked highly, although it's nice to see that happen; I am here to create a permanent resource and figure out new ideas. While I'm also interested in seeing the Lab run effectively, trying to outsmart the search engines is a never-ending battle that I don't think that we should begin. We should focus on how useful we are for our users (including ourselves). That includes ASCII page titles now (in my opinion), but perhaps not forever.

As for ranking within the Lab: as Eric said below, Google will adjust if we change our links; it knows what it is doing.

Posted by: Toby Bartels on May 26, 2009 12:14 AM | Permalink | Reply to this

Re: nLab - More General Discussion

By the way, I believe that google ranks pages by the number of links pointing to them. Currently, google ranks the ASCII titles higher because there are more links to these pages. If we ever did go through and change everything to symbols, then google would eventually pick up the change and the ranking would be as desired.

Posted by: Eric on May 24, 2009 3:59 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Surely google looks at the statistics of links and of clicking on offered search results; however I do believe that it does put the weight on term finds in the title. For example, I tried omega-category. All FIRST 5 entries from nlab are with omega category in the title: omega-category in nLab, simplicial model for weak omega-categories in nLab, discussion on terminology – omega-category in nLab, weak omega-category in nLab, strict omega-category in nLab. Then after the 5, other nlab things appear including late on place 9 infinity-category in nlab. People looking for omega-category certainly are less likely to look for “discussion on terminology – omega-category in nlab”, but because omega-category is in the title google thinks it is more relevant for the search for omega-category then infinity-category in nlab.

I do not understand Toby’s argument that he does care to focus on nlab end-users, but somehow does not consider crucial to this good indexing ? I do not consider that trying to optimize is vanity, on the contrary, failing to think cold-blodedly on optimization is closer there; rights-and-duties arguments like “it is google’s job to optimise”. I find a subjective belief in some big scheme of thing (please do not get insulted by this remark, but I find this kind of beliefs typically American), reminding of usual dogmas of modern society like “democracy”. GM “duty” this way would be to make efficient cars, and it badly failed to do this until it was forced by Japanese. I do not believe that people/sites like us make revenues to google so forget about their technical care on this kind of sites like ours. On the other hand, my own experience is that most users not associated to us personally, and about whose usage I learned accidentaly, are mainly those who came there lead by search engines. So why to think it is dangerous to rely on this (as it looks to me) basic fact ?

I surely agree with Toby that in far future emphasis on ascii will be less relevant; though non-symbols will probably stay most robust kind of data.

By the way, I find the main present deficiency of the nlab to be so much emphasis on numerous variants of definitions while we rarely write proofs of basic facts, what would be the most useful. If we envision much more theorems and proofs in future, this will require some thought in planning the distribution among pages, and title planning.

My desktop was banned to post this post (linux redhat 9 with old version of opera browser) though I can preview the posts. So I needed to ftp to my windows laptop as a number of times before, in order to post.

Posted by: Zoran Skoda on May 26, 2009 9:02 PM | Permalink | Reply to this

Re: nLab - More General Discussion

I find the main present deficiency of the nlab to be so much emphasis on numerous variants of definitions while we rarely write proofs of basic facts

Yes, but the only way out is to start doing it oneself.

For what it’s worth, more recently I started putting in more definition-lemma-theorem-proof material.

Notably at [[geometric embedding]] where the relation to localization is described, and then further at [[sheaf]] where the goal is to give a derivation of Grothendieck topology and the corresponding sheaves starting just with the data of a geometric embedding into a presheaf category.

I’d be grateful for critical reading of this material.

Posted by: Urs Schreiber on May 27, 2009 10:14 AM | Permalink | Reply to this

Re: special symbols in entry names

For example, I tried omega-category. All FIRST 5 entries from nlab are with omega category in the title: […]

If you limit your argument to ranking within the Lab, then I don't mind. You began with ‘I want to see nlab entries high in the google search, this makes our effort more useful.’, and that is what I object to. More on that below, but if you're only interested now in ranking within the Lab, then I agree that Google's behaviour is a valid criterion. (I do think that Google should realise that the words ‘ω’ and ‘omega’ mean the same thing, and I expect that someday it will, but it hasn't yet.)

“it is google’s job to optimise”.

I never said that! It is Google's job to rank pages, and it also Google's job to thwart attempts by web pages to optimise their rankings. It's my understanding that Google does that job well, and I don't want to get involved in that battle; not only is it unethical, but we'd lose. (But this is only relevant to getting our site placed above other sites, not to ranking within our own site.)

(please do not get insulted by this remark, but I find this kind of beliefs typically American)

Why mention that if you don't want people to get insulted by it? Shall I mention that I find the attempt to get automated systems to value one's content more highly typically non-American, on the grounds that most of the spam that passes through my email account (and a higher proportion than the non-spam) is non-American? But that has nothing to do with you, nor would your nationality be relevant even if you did have something to do with it.

By the way, I find the main present deficiency of the nlab to be so much emphasis on numerous variants of definitions while we rarely write proofs of basic facts, what would be the most useful. If we envision much more theorems and proofs in future, this will require some thought in planning the distribution among pages, and title planning.

That's a good point; perhaps you should create a new thread about that here or on the Forum.

My desktop was banned to post this post (linux redhat 9 with old version of opera browser) though I can preview the posts. So I needed to ftp to my windows laptop as a number of times before, in order to post.

Do you get the message described here? This is a result of the overzealous spam-fighting that Jacques has programmed into the Café; if you click on the wrong Post link (which should only happen if you have Javascript turned off, in which case there should be a warning), then your IP address (or something like that) gets banned. Jacques can unban you, and I think that John, Urs, and David can unban you too; you should probably email one of them.

Posted by: Toby Bartels on May 27, 2009 1:44 AM | Permalink | Reply to this

Re: special symbols in entry names

I think that John, Urs, and David can unban you too

Alas, we can’t!

I have myself run into this problem and was not allowed to post here in the past. For me it always went away after a while. If the problem persists for somebody, I know of no other way than to kindly ask Jacques Distler to help.

Posted by: Urs Schreiber on May 27, 2009 10:08 AM | Permalink | Reply to this

Re: special symbols in entry names

Toby wrote: “You began with ‘I want to see nlab entries high in the google search, this makes our effort more useful’, and that is what I object to.”

This incorrect phrasing is indeed my fault, though it should be clear from my post to latest-changes in nlab that I primarily meant “relevant nlab entries” (as you put it standing within the lab) – relative standing of relevant entries to those less relevant. I do not consider putting nlab entries hi in general, particularly those not-relevant hi in comparison to other kinds of search results, generally valuable strategy (I am not sure if my reason is the same as yours however). Namelly, it is good that mathematicians do have some peace and isolation; noise and spam-inducing factors should be eventually avoided. In the cases when there are similar webpages like wikipedia entry etc. I however still do care that better quality entries which precisely match the search are higher (even not only within the nlab). In number of areas, where our nlab is less focused, existing wikipedia entries are better; while in most of the cases of our central interests (category theory proper for example) many nlab entries are now already better.

> Why mention that if you don’t want people to get insulted by it?

Dear Toby, because I believe that the statement is statistically true and that adults appreciate honest information; in my almost 10 years in US I did not have a single social day without hearing statements on obligations of all entities from people to institutions in the mechanics of the System; the thing which I happily most of the time forget even to exist since I left. Of course a very intelligent and independent person as I think you are, may at first feel it extremely unlikely to be influenced by common default patterns of thinking as well, so my comparison-warning that it may be so in this particular case, is just a wanted trigger for your critical self-reflection which may be only useful.

Though I appreciate your semantic correction, I even somewhat disagree that “google’s job is to rank pages” etc. We do not pay google to do anything for us. They have a business model to get money through advertising (mainly paid search-result links). We are just a third party which consumes paid advertising during when ‘using’ their search results. Google undoubtely does have some sophisticed quality control of their search results also toward us (I mean quality of satisfying our search wishes), in order to stay number one global general search machine, but there is no value, moral or obligation statement on the content of their “job” there. If they find any change of goal in structuring their web page outputs with commercials, search results, ranking lists, excerpts, image samples etc. they will change. The model of google’s inner works where an efficient ranking is the only hi priority is just a model, more or less successul (while depending crucially on their current model/definition of efficiency). NOTHING is google’s job except making money. Leo Tolstoy used to say for people that the only thing a person ultimately needs is a six feet long place in the dust to die. Everything else is propaganda or prejudice.

As far as banning message, it was
“You are not allowed to post comments.” in a phase after I successfully done the preview (and the browser automatically filled my personal data in the form).

Posted by: Zoran Skoda on May 27, 2009 2:58 PM | Permalink | Reply to this

Category of Elements

I’m trying to understand category of elements.

First question, why is it defined with a function P:C opSetP:C^{op}\to Set instead of a functor P:CSetP:C\to Set? The latter would make more sense to me. I was tempted to change the definition so that we define it in terms of concrete categories. Is there a compelling reason to define it in terms of presheafs instead?

Posted by: Eric on May 28, 2009 10:43 PM | Permalink | Reply to this

Re: Category of Elements

Because a category of elements is defined with a functor P:C opSetP:C^{op}\to Set instead of a functor P:CSetP:C\to Set, it seems like my illustration is wrong. I’d need to reverse the arrows in the bottom figure (unless I’m confused).

Could we just change the definition so it uses P:CSetP: C\to Set? Then my diagram would be correct (I think!). Also, 2 of the three equivalences need to invoke its “opposite”. If we define category of element with P:CSetP:C\to Set, I think only the first would require invoking the opposite.

Posted by: Eric on May 28, 2009 11:03 PM | Permalink | Reply to this

Re: Category of Elements

Could we just change the definition

We could. But then, in other contexts one wants it the other way round.

So more importantly, whichever way we end up stating the definition, is that we insert the remark that it is not relevant for the general concept.

Posted by: Urs Schreiber on May 28, 2009 11:13 PM | Permalink | Reply to this

Re: Category of Elements

I should rather ask…

Would there be any objection to changing it?

The concept has a nice picture that I think is worth maintaining. Sticking an unnecessary “op” in there messes up the picture.

Plus, Grothendieck construction uses F:CSetF:C\to Set.

I’m leaning toward changing it…

Then, would it be restrictive in any way to define it in terms of concrete categories (requiring the functor F:CSetF:C\to Set to be full faithful and representable)?

Posted by: Eric on May 28, 2009 11:39 PM | Permalink | Reply to this

Re: Category of Elements

why is it defined with a functot P:C opSetP : C^{op} \to Set instead of a functor P:CSetP : C \to Set?

This has no intrinsic relevance. One is as good as the other. It’s just a matter of convention. You don’t need to worry about this, you can just as well think of P:CSetP : C \to Set throughout.

One uses functors out of the opposite category if one wants to think of the collection [C op,Set][C^{op}, Set] of all such as aa generalization of CC itself. It turns out that to do so, there is this little op{}^{op} required. With it we get an injection

C[C op,Set] C \hookrightarrow [C^{op}, Set]

(called, by the way, the [[Yoneda embedding]]) and this injection is something very useful in many circumstances.

HOWEVER, this injection is pretty irrelevant in the context in which you are looking currently at the notion “category of elements”. So you can completely ignore it and think of functors P:CSetP : C \to Set.

(Of course there is conversely an embedding C op[C,Set]C^{op} \to [C,Set], but that, too, is not to be worried about here).

So all you need to notice is that every category is the opposite of some category, namely of its opposite,

C=(C op) op C = (C^{op})^{op}

so that whenever you see a C opC^{op} somewhere and don’t like it, you can consider the case that C=D opC = D^{op} and replace all C opC^{op}s by DDs.

Posted by: Urs Schreiber on May 28, 2009 11:10 PM | Permalink | Reply to this

Re: Category of Elements

I think (or rather guess), that the preference for [𝒞 op,Set][\mathcal{C}^{op},\mathrm{Set}] over [𝒞,Set][\mathcal{C},\mathrm{Set}] is motivated by situations where the 𝒞\mathcal{C} in question occurs as a (small) subcategory of some category 𝒳\mathcal{X} and where there is already another notion of (generalized) element for objects of 𝒳\mathcal{X}. Here an element x TXx\in_T X is just a map x:TXx: T \rightarrow X with codomain XX. In particular one can restrict attention to 𝒞\mathcal{C}-elements of XX, i.e. the x CXx\in_C X for CC from 𝒞\mathcal{C}.

Then the category of elements for the functor 𝒳(,X):𝒞 opSet\mathcal{X}(-,X): \mathcal{C}^{op} \rightarrow \mathrm{Set} is the same as the category 𝒞/X\mathcal{C}/X of 𝒞\mathcal{C}-elements of XX.

Of course, one could just dualize and replace 𝒞\mathcal{C} and 𝒳\mathcal{X} with 𝒞 op\mathcal{C}^{op} and 𝒳 op\mathcal{X}^{op}. In terms of the original 𝒳\mathcal{X} this means considering functors 𝒳(X,):𝒞Set\mathcal{X}(X,-): \mathcal{C} \rightarrow \mathrm{Set}. But maps XCX\rightarrow C are more like functions and not elements.

Posted by: Marc Olschok on May 30, 2009 11:17 PM | Permalink | Reply to this

Re: Category of Elements

Don’t forget that the first examples of generalised elements were local sections of sheaves on spaces where the category is the opposite of the category of open sets (due to the operation of morphisms (which are the inclusions) is precisely restriction of local sections. That is where the historical preference for 𝒞 op\mathcal{C}^{op} came from. It also is a very good intuition for handling generalise elements in general (that is provided you like the intuitions of local sections of a bundle or similar thing).
Posted by: Tim Porter on May 31, 2009 8:15 AM | Permalink | Reply to this

Re: Category of Elements

Hey Tim, how'd you manage to post invalid XHTML? (no <p> tag). Don't worry about it if you don't know; I'll send Jacques an email. (Actually, you may be able to tell from this post that I figured it out.)
Posted by: Toby Bartels on May 31, 2009 6:08 PM | Permalink | Reply to this

Image Sizes

I’ve uploaded a jpeg. Is there a way to modify the size of the displayed image from within the link?

Posted by: Eric on May 29, 2009 6:56 PM | Permalink | Reply to this

Re: Image Sizes

Is there a way to modify the size of the displayed image from within the link?

Eric, that question should go to the forum. Please take it there.

(I don’t know the answer to your question except for: did you look at the code surrounding the picture inclusion on nnLab pages such as this. Doesn’t that set the size somehow, too? Maybe not.)

Posted by: Urs Schreiber on May 29, 2009 10:08 PM | Permalink | Reply to this

Re: Image Sizes

Thanks.

I guess the trick is to

1. Upload the figure
2. Use old fashioned html

I didn’t realize the forum was active. I’ve subscribed to the feed now and will direct questions like this there from now (I suppose).

Posted by: Eric on May 29, 2009 11:01 PM | Permalink | Reply to this

Suggestions for a name

Does anyone have suggestions for a name for the following concept: an nn-category internal to kk-categories, satisfying fibrational conditions which ensure among other things that it has an underlying (n+k)(n+k)-category?

When n=k=1n=k=1 this is what I’ve called a “framed bicategory,” but that name doesn’t really scale well to larger values of nn and kk, when it starts to feel even less like an (n+k)(n+k)-category with extra structure. Not to mention that whenever I talk about them, almost without fail someone asks me why “bicategory” when they are really a type of double category. Also, in retrospect “framed” is an unfortunate adjective, because these things seem to be coming up a lot in topological field theory, where “framed” is applied to manifolds with a completely different meaning. For instance, at the TFT conference Chris Douglas discussed an example with n=2n=2, k=1k=1 (conformal nets, algebras, and modules) and also one for n=1n=1, k=2k=2 (monoidal categories and their category-modules).

I think the best I have come up with so far is (n×k)(n\times k)-category (pronounced “nn by kk category”). It’s concise and it suggests that the nn and kk are the dimensions of a rectangle, which is almost right: an (n×k)(n\times k)-category has (n+1)(k+1)(n+1)(k+1) types of cells which are arranged naturally in a rectangle. And it also generalizes in the other dimension, e.g. a 11x11x11-category would be a certain sort of triple category (a category in categories in Cat). Reactions?

Posted by: Mike Shulman on June 2, 2009 2:58 AM | Permalink | Reply to this

Re: Suggestions for a name

I like it.

I wouldn't worry too much about the numbering; it's just further evidence that the numbering for nn-categories if off by 11 from the natural numbering, but that's endemic and anything done to fix it later will apply to your concept just as easily.

My only concern is that one might want a general term for an nn-category in kCatk Cat without the fibrational conditions, and ‘nn-by-kk-category’ is already a very natural term for that. (In particular, then an nn-by-nn-category will be a double nn-category; that's nice, isn't it?) Any chance that there's a good adjective to stick in front of ‘nn-by-kk-category’ here? or a good argument to be made that framed bicategories are really a more natural concept than double categories anyway?

(By the way, I write ‘mm-by-nn matrix’ instead of ‘m×nm \times n matrix’ too, so don't pay any attention to that.)

Posted by: Toby Bartels on June 2, 2009 6:25 PM | Permalink | Reply to this

Re: Suggestions for a name

Has no one else noticed the dangers of m by n matrices, especially with people who mumble or have sloppy handwriting on the board like mine?

Did no one else grow up being told to mind thier p’s and q’s? :-)

Posted by: jim stasheff on June 2, 2009 7:50 PM | Permalink | Reply to this

Re: Suggestions for a name

I wasn’t really worried about the numbering being off; as you say it’s just the same way that “nn-category” is off.

Your concern about nn-categories in kCatk Cat without the fibrational condition occurred to me as well. The best adjective I can think of right now to distinguish these is “fibered.” This isn’t ideal, though, since “fibered nn-category” has a quite different meaning.

However, I can’t think of any non-fibered nn-categories in kCatk Cat for n>1n\gt 1 or k>1k\gt 1. It’s a funny thing: I can think of three flavors of double categories. The first are framed bicategories, which seem to generalize most naturally to (fibered) (n×k)(n\times k)-categories. The second are the ones used by Brown/Loday/et.al. to model homotopy types, and are usually generalized to nn-fold categories (i.e. non-fibered 1x1x\dotsx1-categories)—but perhaps they could be generalized further to non-fibered (k 1××k m)(k_1\times\dots\times k_m)-categories.

The third are double categories like (monoidal categories, oplax functors, and lax functors) or (model categories, left Quillen functors, and right Quillen functors), in which the vertical and horizontal morphisms are “symmetric” and “dual.” A higher-dimensional generalization of these seems much less straightforward and will probably remain fundamentally “double” in some way.

Perhaps for conceptual clarity it would be best to add an adjective like “fibered.” I would then be tempted to write “all (n×k)(n\times k)-categories will be assumed to be fibered” at the beginning of a paper.

Posted by: Mike Shulman on June 2, 2009 8:24 PM | Permalink | Reply to this

Re: Suggestions for a name

This isn’t ideal, though, since “fibered nn-category” has a quite different meaning.

How about ‘fibrational’? Is that taken too?, or just too ugly?.

Posted by: Toby Bartels on June 2, 2009 11:20 PM | Permalink | Reply to this

Re: Suggestions for a name

How about ‘fibrational’? Is that taken too?, or just too ugly?

Kind of ugly, I think. One could say “fibrant,” although that might have its own problems.

Posted by: Mike Shulman on June 3, 2009 3:03 AM | Permalink | Reply to this

Re: Suggestions for a name

“Fibrous”? Or would the multiplication of different “fibre”-related words get too difficult to keep track of?

Posted by: Tim Silverman on June 3, 2009 6:24 PM | Permalink | Reply to this

Redirects!

There is a new redirect feature on the Lab. Don't push it too hard yet, but try it out in the Sandbox, new test pages, and maybe some current redirects that you know have no real content even in their edit histories.

Jacques's explanation (and subsequent complaints/praise) may be found here.

Ongoing discussion (both before and after the new feature appeared) without Jacques may be found here.

Posted by: Toby Bartels on June 3, 2009 8:30 PM | Permalink | Reply to this

Re: Redirects!

Please keep the discussion on the forum (i.e. not on Jacques’ blog). As it’s a new feature, it may take a while before we really understand how it works and I think it’s best if we keep that discussion amongst ourselves.

If there are any changes we’d like then we should come to a consensus before asking Jacques to implement them as we may not all want the same changes.

Posted by: Andrew Stacey on June 4, 2009 8:19 AM | Permalink | Reply to this

server

We are facing the following technical problem:

the nnLab’s “recently revised”-page has become effectively unavailable. Calling it seems to send the server down a long road that it ever more rarely finds its way back from.

On the weekend or else next week Andrew Stacey, Toby Bartels and myself will try to migrate the nnLab to another hosting company in the hope that a less sluggish server will do us good.

Meanwhile I’d ask everyone

- don’t call “recently revised” – it hurts the performance for all of us

- but all the more important: leave a note (ever so brief) at latest changes whenever you do anything on the Lab.

Hope things will get better soon. Meanwhile, it tuns out that without calling “recently revised” the responsiveness of the machine is actually pretty decent.

By the way, John is now starting to explain stuff on the nnLab, the Baezian style. Check out [[free cocompletion]]!

Posted by: Urs Schreiber on July 16, 2009 10:05 PM | Permalink | Reply to this

Re: nLab - More General Discussion

I am a bit suspicious that moving to a new server will make the problem long term solved. It may be that some of the way the software is setup are inherently inefficient; I heard that it is rather often when working with Ruby that people need to tweak for performance.

Posted by: Zoran Skoda on July 18, 2009 4:18 PM | Permalink | Reply to this

Re: nLab - More General Discussion

That’s as maybe. You may be right, but the set-up in the moment is not optimal. For example, we’re using the built-in database (sqlite3) rather than farming that off to a separate MySQL server. The operations that have been identified as slowing down the n-lab are those that involve parsing the entire database and thus are quite likely to be improved by improving the underlying database.

Changing server isn’t just about improving performance. The current server has other annoyances that mean that a change is desired. As this will hopefully happen soon, it’s not worth trying to tweak the performance on the current set-up but rather do this as the system is shifted over. Thus it’s “changing the server will make things better” but rather “changing the server will make it easier for us to make things better”.

Posted by: Andrew Stacey on July 19, 2009 9:51 PM | Permalink | Reply to this

Re: nLab - More General Discussion

The Lab is down, and I can't get it to restart.

If I log in to the server and restart lighttpd, then that goes fine but doesn't fix things.

If I try to restart instiki, then it fails to stop it (no PID file) and claims that it's already running.

But if I look at ps -A, instiki is not there.

Fortunately, I saved my work in an external editor. (^_^)

Posted by: Toby Bartels on August 26, 2009 6:05 PM | Permalink | Reply to this

Re: nLab - More General Discussion

It's back now, so if somebody did something, please explain it, so I could do it too!

Posted by: Toby Bartels on August 26, 2009 7:48 PM | Permalink | Reply to this

Re: nLab - More General Discussion

Down again.

It really has been behaving especially badly today, and I'm pretty sure that I'm the only one using it for the past few hours. The change in behaviour of instiki restart is also troubling. Has something been changed in the server's configuration?

See also http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=56.

Posted by: Toby Bartels on August 26, 2009 9:20 PM | Permalink | Reply to this

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