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December 2, 2010

Category Theory Puzzle

Posted by John Baez

One nice thing about category theory is that despite its soaringly ambitious nature, it still contains hundreds of satisfying little puzzles to entertain the problem-solver in us. James Dolan likes to give me these puzzles and see how long it takes me to solve them. While I find it a bit distressing to be put on the spot like that, they’re still fun.

Here’s the one he gave me yesterday. See how long it takes you.

Suppose CC is an AbAb-enriched category: that is, its homsets are abelian groups and composition is bilinear. Suppose xCx \in C is a terminal object. Show that xx is initial.

Posted at December 2, 2010 3:21 AM UTC

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Re: Category Theory Puzzle

Good puzzle! Do I detect Jim’s hand in calling an object of a category “xx”?

Maybe you should tell us who’s allowed to post an answer.

Posted by: Tom Leinster on December 2, 2010 4:06 AM | Permalink | Reply to this

Re: Category Theory Puzzle

I guess Jim was the one who helped me relax and use lower-case for objects of categories. If elements look like xx, and sets look like XX, and categories look like X, by the time we get to 2- or 3-categories the letters will get so big and scary they’ll make us run out of the room screaming!

But Jim just uses the same font for everything, where I like to use a couple.

Maybe you can start by saying how long it took for you to do this puzzle, or if you’d already known it.

Posted by: John Baez on December 2, 2010 4:22 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Hmm… I don’t really like the idea of comparing times. But to answer your other question, I’m pretty sure I’d done it before, probably as part of a problem sheet from Peter Johnstone.

I knew it as one aspect of the fact that if you’re a category enriched in abelian groups (or just commutative monoids) then you have finite products if and only if you have finite coproducts—and in that case they’re the same. That’s another good puzzle, though not as neat as this one.

Gbz

Posted by: Tom Leinster on December 2, 2010 5:19 AM | Permalink | Reply to this

Re: Category Theory Puzzle

I agree that it’s sort of tacky to say how long it took to solve this puzzle — but I think it’s fun as long as one doesn’t get too competitive about it. The answer for me is somewhere between 10 minutes and 15 hours, depending on how you count. I don’t think that’s setting too high a bar.

I noticed that it worked just as well for commutative monoids.

It was on Jim’s agenda to check that finite products in an Ab-enriched category are automatically finite coproducts, so it’s nice to hear that’s true.

(All this is a small part of some grandiose scheme of Jim’s, which he is explaining to me.)

(By the way: if everyone writes using rot13, people may claim that proofs in category theory are just a bunch of gobbledygook.)

(Also by the way: I wonder how many people have learned to read rot13. Jim says he used to be able to…)

Posted by: John Baez on December 2, 2010 7:11 AM | Permalink | Reply to this

Re: Category Theory Puzzle

I would like to know what is really going on with proofs of this kind. At the next level up, in a bicategory whose homs are cocomplete, and in which those colimits distribute over composition (e.g., the bicategory of profunctors), all bicolimits are also bilimits. One imagines there must be good general reasons why such results are true.

Posted by: Richard Garner on December 2, 2010 7:44 AM | Permalink | Reply to this

Re: Category Theory Puzzle

I would like to know what is really going on with proofs of this kind. At the next level up, in a bicategory whose homs are cocomplete, and in which those colimits distribute over composition (e.g., the bicategory of profunctors), all bicolimits are also bilimits. One imagines there must be good general reasons why such results are true.

my (possibly hallucinated) “grandiose scheme” (which as usual is probably just a reinvention of ideas already known within some mathematical subculture, but is intended to be more humanly accessible) involves another example where “weak colimits and weak limits are almost the same thing”, namely when the hom objects are chain complexes, or more generally modules of some e^infinity ring. i’m somewhat surprised by your example because i was imagining that my example was rather special; i should think about your example and how it might relate to mine. it reminds me of things that todd trimble has tried to teach me about in the past about adjoint bimodules and related stuff.

Posted by: james dolan on December 2, 2010 11:46 AM | Permalink | Reply to this

Re: Category Theory Puzzle

namely when the hom objects are chain complexes, or more generally modules of some E E_\infty ring.

Specifically with enrichement over spectra we have exactly the same reasoning as with enrichment over abelian groups: since the tensor product is a smash product, 0-morphisms are absorbing.

Posted by: Urs Schreiber on December 2, 2010 12:06 PM | Permalink | Reply to this

Re: Category Theory Puzzle

Specifically with enrichement over spectra we have exactly the same reasoning as with enrichment over abelian groups:

the result in question isn’t true for abelian groups. a kernel in an algebroid isn’t a weighted colimit, and a cokernel in an algebroid isn’t a weighted limit, at least not in the sense that i am (and i think also richard garner is) talking about.

Posted by: james dolan on December 2, 2010 12:50 PM | Permalink | Reply to this

Re: Category Theory Puzzle

the result in question isn’t true for abelian groups

I mean the statement that a terminal object is necessarily also initial. Wasn’t clear to me that you switched to the topic of bilimits coinciding with bicolimits. I thought you were just invoking that to say something about zero objects.

Posted by: Urs Schreiber on December 2, 2010 1:40 PM | Permalink | Reply to this

Re: Category Theory Puzzle

I view these result as special cases of the general fact that for any enriching category VV, Cauchy colimits are the same as co-Cauchy limits. More precisely, if a profunctor HH has a right adjoint KK, then HH-weighted colimits are the same as KK-weighted limits. If you take V=V= pointed sets and write down the weight HH for initial objects, you’ll see that it has a right adjoint which is the weight for terminal objects. Similarly for finite products and coproducts when V=V= abelian monoids, and for arbitrary products and coproducts when V=V= suplattices. I suspect that when VV is the 2-category of cocomplete categories with its “bilinear” tensor product, there is a similar adjointness of 2-profunctors.

Posted by: Mike Shulman on December 2, 2010 7:23 PM | Permalink | Reply to this

Re: Category Theory Puzzle

yes, that’s the way i’ve been thinking about it too, though since i’ve been re-inventing it for myself i don’t know the standard terminology very well and i’m also vague on a lot of the technical details.

i’m interested in the general question of when does an enriching category (or higher category) v have the property that all (or “almost all”) pro-v-functors have adjoints? for example is there some interesting sense in which you could start with v = _finset_ and then universally compel all pro-v-functors to have adjoints (perhaps while maintaining some other desirable properties), and what would that universal example be like?

i didn’t even know about the example of v = _cocomplete semilattice_ until richard garner pointed it out. the example that i’ve been thinking about is v = _r-module_ (considered as a symmetric monoidal (infinity,1)-category, or something like that) where r is an e^infinity ring; or as a special case of that, _dg r-module_ where r is a differential graded-commutative ring.

in this example it seems unlikely that literally all pro-v-functors can have adjoints, but my guesses and some calculations suggest that any “reasonable” pro-v-functor has a (weak) adjoint. (i think that “reasonableness” here mainly involves being finitely presented in some appropriate sense.)

the intuition behind this is based in part on the slogan that homological algebra and derived functors are all about “restoration of exactness”, or more generally that stable homotopy theory is about unifying the concepts of “fibration sequence” and “cofibration sequence”. (and also another slogan that simon willerton tried to explain to me, that derived categories were invented for the purpose of getting “duality” to work better; but don’t blame simon if i’m mangling the slogan.)

clearly a kernel in an algebroid is not a weighted colimit and a cokernel in an algebroid is not a weighted limit, or else any merely half-exact functor would be exact. homological algebra is all about the fact that there are lots of functors that are only half-exact, but also that they become homotopy-exact when you pass to the total derived functor. the homotopy-kernel of a chain map _is_ a weighted homotopy-colimit and its homotopy-cokernel _is_ a weighted homotopy-limit, because the homotopy-kernel and the homotopy-cokernel are really the same thing, just shifted by 3 steps in the unified fibration/cofibration sequence of the chain map.

this “restoration of exactness”, the fact that cauchyness spreads from merely the finite discrete colimits to much more general (homotopy-)colimits when you pass from abelian groups to (“2-sided”) chain complexes, means that the (weak) cauchy completion of a dg algebroid is almost the same thing as either its free completion or its free cocompletion. peter freyd (i think) showed that the “free abelian category” monad on the 2-category of algebroids is a composite of 2 monads, one that freely adjoins finite limits and one that freely adjoins finite colimits, with an invertible barr-beck distributivity-braiding natural transformation mediating between them. but in the chain complex world these two monads merge into a single idempotent monad, and the braiding between them degenerates into the identity transformation.

my interest in trying to understand these ideas has been motivated in large part by my interest in trying to understand algebraic geometry. i know that a nice algebraic “stack” or “scheme” can be construed as the moduli stack of models of a “theory” expressed in the “doctrine” of symmetric monoidal finitely cocomplete algebroids; the objects of the theory are known as “coherent sheaves”. (see here. for example, the coherent sheaves over the moduli stack of n-dimensional lie algebras form the free symmetric monoidal finitely cocomplete algebroid on one n-dimensional lie object, and the coherent sheaves over projective n-space form the free symmetric monoidal finitely cocomplete algebroid on a 1-dimensional object equipped with an embedding into standard linear [n+1]-space, and so forth.) but the special emphasis on finite colimits as opposed to finite limits here has bugged me for a long time. but if you pass from the doctrine of symmetric monoidal finitely cocomplete algebroids to the “higher doctrine” of symmetric monoidal finitely homotopy-cocomplete dg algebroids (or something like that) by passing from the coherent sheaves to the chain complexes of coherent sheaves, then it seems that the cauchyness of the homotopy-colimits in some sense corrects the imbalance between the limits and the colimits.

i’d like to try to understand things like “serre duality” and “verdier duality” and “grothendieck’s six operations” (and other stuff that simon has tried to explain to me) from this viewpoint of trying to exploit the cauchyness of homotopy-colimits of chain complexes. at least some of these things have to do with adjoint bimodules that exist at the level of derived categories but not at the level of the original abelian categories.

in general, the “inverse image” functor of a morphism between algebraic stacks or schemes is only half-exact, preserving finite colimits of coherent sheaves (as well as their tensor products) but not finite limits. when it does preserve finite limits then the morphism is said to be “flat”, which is a strikingly geometric name for such a seemingly algebraic property. the geometric interpretation of flatness is similar to what it sounds like, that in some sense the geometric fibers of the morphism vary in a way that’s not too wild, though how the algebraic property leads to this geometric interpretation is still somewhat obscure to me. but when we pass to the chain complexes of coherent sheaves then the cauchyness of the homotopy-colimits implies that inverse image is now automatically homotopy-exact. very vaguely, this reminds me of how the property of a continuous map of being a fibration becomes in a sense vacuous at the homotopy level, in that the homotopy-fibers of the map vary in a tame way even if the actual fibers vary wildly. i don’t know whether that’s actually a good intuition about how flatness works, though.

anyway, probably most of what i’m saying here is either well-known somewhere (stuff like “it’s easier to get adjoints of bimodules if you pass to the derived category”), or wrong.

Posted by: james dolan on December 3, 2010 10:51 PM | Permalink | Reply to this

Re: Category Theory Puzzle

my guesses and some calculations suggest that any “reasonable” pro-v-functor has a (weak) adjoint. (i think that “reasonableness” here mainly involves being finitely presented in some appropriate sense.)

I could believe that. For instance, when enriching over plain old R-modules for an ordinary ring R, the dualizable objects are the finitely generated projectives. Projectivity is often a way of saying that something non-homotopical is nevertheless “homotopically well-behaved,” and I could believe that that’s the case here.

And in the ordinary stable homotopy category, I believe that all finite cell objects are dualizable, which is basically a notion of “finite presentation” for spectra. Duality of suspension spectra (of finite CW complexes) in the stable homotopy category is also called Spanier-Whitehead duality (esp. when reformulated in unstable terms). And at the end of chapter 7 of EKMM there is an extension of that to R-modules for any commutative S-algebra (= E E_\infty-ring) R.

Posted by: Mike Shulman on December 4, 2010 4:06 AM | Permalink | Reply to this

Re: Category Theory Puzzle

do you think that there might be any interesting sense in which richard garner’s example of v = _cocomplete category_ might be the “universal” way to provide adjoints for pro-v-functors if the “adjoints” are allowed to be of a certain “weak” (maybe “lax” in some sense, or maybe just “2-weak”?) kind, while v = _spectrum_ might be the universal way if they’re allowed to be weak in a certain different way (maybe “(infinity,1)-weak” or something?)?

this is just a vague silly guess…

do the adjoints for pro-v-functors where v = _cocomplete poset_ enjoy some stronger formal property than the ones for where v = _cocomplete category_?

i still haven’t really understood these other two examples (v = _cocomplete poset_ and v = _cocomplete category_) in any detail yet so i could be off on the wrong track here.

Posted by: james dolan on December 4, 2010 4:51 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Those are interesting questions; I don’t know the answers right now. But you are right, I think, that the “adjoints” in the cocomplete-category case are kind of lax. I was thinking last night about how you would construct that adjunction, and it seemed to me that you would need both (a) morphisms of 2-profunctors to be lax natural transformations, and (b) the “tensor product” of 2-profunctors to be a lax colimit. I’m not sure in what formal context one could make sense of that (e.g. is “composition” of 2-profunctors via lax colimits associative, unital, and functorial on lax natural transformations?).

Posted by: Mike Shulman on December 4, 2010 4:49 PM | Permalink | Reply to this

Re: Category Theory Puzzle

my interest in trying to understand these ideas has been motivated in large part by my interest in trying to understand algebraic geometry. i know that a nice algebraic “stack” or “scheme” can be construed as the moduli stack of models of a “theory” expressed in the “doctrine” of symmetric monoidal finitely cocomplete algebroids; the objects of the theory are known as “coherent sheaves”.

I am wondering how this is related to the following general abstract conception of coherent sheaves of modules:

For CAlg opC \subset Alg^{op} the \infty-site of formal duals of \infty-algebras in question, the \infty-stack of quasicoherent \infty-stacks is the almost-tautological one that assigns SpecAAModSpec A \mapsto A Mod.

In order to make explicit the general abstract origin of this characterization, one observes that there is a generalization of Quillen’s old characterization RModAb(Ring/R)R Mod \simeq Ab(Ring/R) of the category of modules over a rings as the category of abelian group objects in the overcategory of all rings over the given one. In fact the category ModMod of modules over all possible rings is the fiberwise abelianization of the codomain fibration over RingRing: the tangent category TRingRingT Ring \to Ring. This is the stack of modules in 1-category theory, regarded under Grothendieck construction as a cofibration.

Analogously, for CC any \infty-site, there is the corresponding tangent \infty-category TC opC opT C^{op} \to C^{op}, given by fiberwise stabilization of the codomain \infty-fibration. This is the fibration incarnation of the \infty-stack of quasicoherent \infty-stacks.

Specifically for CdgAlg opC \subset dgAlg^{op} this gives complexes of quasicoherent sheaves.

There is possibly an even more general abstract way to say this. And maybe that’s what you are getting at?

Posted by: Urs Schreiber on December 4, 2010 10:39 AM | Permalink | Reply to this

Re: Category Theory Puzzle

i haven’t yet been able to understand what you wrote well enough for me to answer your question. which isn’t surprising; i often have lots of trouble understanding mathematical writing.

to try to see whether we can figure out what each other is talking about, it might be a good idea to mostly stay away from higher categories at first. in the paragraph of mine that you quoted, i’m pretty much only talking about a single groupoid-enriched category of structured 1-categories.

(i need a convenient short term here for “groupoid-enriched category” so i’ll use “g-category” for this purpose.)

the g-category that i’m talking about is the g-category of symmetric monoidal finitely cocomplete small k-algebroids over some fixed commutative ring k; are you reasonably satisfied for the moment that we both agree what that means?

(i need a convenient short name for this g-category so for the moment i’ll call it “x”.)

one of my main reasons for being interested in x is that it has a substantial overlap with certain g-categories that algebraic geometers study, such as the g-category of “deligne-mumford stacks”. (that is, there are pretty big full sub-g-categories of x that are equivalent to pretty big full sub-g-categories of those other g-categories.)

however, i’m not too interested in the fact that x-objects can be viewed as “stacks” (in the sense of “categorified sheaves”) over some “site”. for one thing, the objects of _any_ g-category can be viewed as stacks over some (higher) site, by turning the yoneda embedding crank. thus, viewing objects of a g-category as stacks of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. but there’s already a very good way of thinking about what x-objects are: they’re symmetric monoidal finitely cocomplete algebroids.

similarly, the objects of any category can be viewed as sheaves over some site by turning the yoneda embedding crank, and thus viewing objects of some category as sheaves of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. thus for example “coherent sheaves” can indeed be thought of as sheaves of a sort but in general there are much better ways of thinking about them.

so even though a lot of the things that i’m studying have traditional names (like “coherent sheaf” or “deligne-mumford stack”) that suggest a connection to the study of sites and sheaves and stacks and so forth, those names are somewhat misleading, mostly just unfortunate historical accidents. i think that this is one of the things that’s making it difficult for me to connect what you wrote (which seems to focus a lot on things like sites and sheaves and stacks and so forth) with what i was talking about. the connection might very well be there; i just didn’t manage to find it yet.

so, one of the ideas that i was talking about is the idea that under good conditions, an “algebraic stack” (such as a “deligne-mumford stack”) can be recovered from its symmetric monoidal finitely cocomplete algebroid of coherent sheaves. this is a very standard idea; for example, in the recent thread here on the “3-fold way”, theo alluded to the fact that the “two point” stack and the “half a point” stack share the same finitely cocomplete algebroid of coherent sheaves, but are distinguished from each other by having different symmetric monoidal structures. being able to recover a stack from an x-object in this way makes the actual stack (and the site where it lives, and so forth) superfluous if we so choose, and i personally often do so choose.

to try to give an idea of how i’m suggesting to think about coherent sheaves instead of thinking of them as sheaves of some sort, it might be helpful to recall a thread here some time ago where tom leinster suggested that the appreciation and exploitation of universal properties of structured categories constitutes a shibboleth that marks a “professional” category-theorist; that although the idea of a universal property has become the property of the wider math community, the special case where the object possessing the universal property is itself a category of some sort has not. in that thread, mike shulman mentioned the relationship of such “categorified universal properties” to the ideas about the use of doctrines in algebraic geometry that i’ve been exploring. but it’s striking that category-theorists aren’t the only mathematical subculture who’ve made an exclusionary shibboleth out of the concept of “categorified universal property”- algebraic geometers speaking their own mathematical dialect have done so too. such a double shibboleth dividing two subcultures constitutes a failure to communicate, an unexploited opportunity to combine the insights of both cultures.

the example that i mentioned of the coherent sheaves over projective n-space as “the free symmetric monoidal finitely cocomplete algebroid on a one-dimensional object equipped with a linear embedding into standard linear [n+1]-space” is an excellent example to work through in detail if you haven’t already done so. algebraic geometers sometimes describe this categorified universal property by saying that projective n-space is “the classifying space for line bundles generated by n+1 sections”.

in trying to understand this categorified universal property, it’s probably more of a hindrance than a help to think of coherent sheaves as sheaves over some site. instead, there’s a way, described in lots of basic algebraic geometry books, to think of coherent sheaves over a projective variety as special graded modules of the “homogeneous coordinate algebra” of the variety (which in the example of projective n-space is just polynomials in n+1 variables). the categorified universal property can be understood very directly from this viewpoint.

category-theorists are perhaps familiar with exploiting the idea of categorified universal properties in the context of “tannaka-krein” theorems, about recovering a group from its representation category; for example, the idea that the finite-dimensional representations of gl(n) form the free symmetric monoidal finitely cocomplete algebroid on one “n-dimensional” object. my impression though is that they’re less familiar with exploiting such categorified universal properties in the context of “gabriel-rosenberg” theorems, a vast generalization of tannaka-krein theorems, about recovering an algebraic stack from its coherent sheaf category. my sense of this estrangement between category-theorists and algebraic geometry is based largely on my own personal experiences, my failure for such a long time to understand what algebraic geometers were saying. i don’t really know for sure whether it’s mainly just me who experienced this estrangement, or whether other category-theorists have had a similar experience.

Posted by: james dolan on December 10, 2010 12:38 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Thanks for the reply.

I will try to answer, but not in detail right now. (It’s 2:30 am, I have to get up early, teach a seminar, then run to catch a train to berlin where I visit my brother over the weekend. So I might be offline for a bit.)

Just a few quick remarks.

(i need a convenient short term here for “groupoid-enriched category” so i’ll use “g-category” for this purpose.)

There is by now a fairly “established” term for this: “(2,1)(2,1)-category”.

the g-category that i’m talking about is the g-category of symmetric monoidal finitely cocomplete small k-algebroids over some fixed commutative ring k; are you reasonably satisfied for the moment that we both agree what that means?

Yes, I know exactly what you mean.

(i need a convenient short name for this g-category so for the moment i’ll call it “xx”.)

I like descriptive terms. It’s like putting books into the shelf in the right order: it facilitates retrieving information without losing time on inessential searches.

Therefore I’d tend to suggest that we call this (2,1)(2,1)-category by a different name, eventually. Something like 2Alg k2Alg_k for instance. This is still not entirely descriptive, as there are various different concepts that can all justly be called a “2-algebra over kk”, but this here is one of them.

At the same time, trying to agree on such a more or less descriptive term may help us get on the same page and find out if we agree on what we are talking about.

however, i’m not too interested in the fact that x-objects can be viewed as “stacks” (in the sense of “categorified sheaves”) over some “site”. for one thing, the objects of any g-category can be viewed as stacks over some (higher) site, by turning the yoneda embedding crank. thus, viewing objects of a g-category as stacks of some sort is something that you might resort to if you can’t find any better way of thinking about those objects. but there’s already a very good way of thinking about what x-objects are: they’re symmetric monoidal finitely cocomplete algebroids.

Sure. The point of identifying something as a stack on something, or better as an object in some (2,1)(2,1)-topos, is not that it can be done at all. The point of it is that is specifies an ambient context in which to reason about the objects under consideration. This becomes relevant as soon as we start considering certain operations on which our objects may not be closed.

There is a differential geometric analog to all that we are discussing here, comparison to which might be useful:

we can talk about orbifolds without thinking of them as stacks on anything. We can think about effective Lie groupoids without thinking of them as stacks on anyting. We can think about general Lie groupoids without thinking of them as stacks on anything…. But at some point, when working with these objects, we will notice that we tend to drop out of the (2,1)(2,1)-categories that they form. At that point, we may feel the need to pass to larger (2,1)(2,1)-categories that contains all them in a useful way. Experience (and Lawvere) indicate that this large ambient (2,1)(2,1)-category should always be taken to be some (2,1)(2,1)-topos. And that will typically have some site of definition, even if we may not care about it.

so, one of the ideas that i was talking about is the idea that under good conditions, an “algebraic stack” (such as a “deligne-mumford stack”) can be recovered from its symmetric monoidal finitely cocomplete algebroid of coherent sheaves. this is a very standard idea;

Yes, as I mentioned in the other thread: this is the idea that notably Kontsevich has been following for more than two decades: describe all things geometric by their collections of quasicoherent sheaves. (Which ultimately always form a stable (,1)(\infty,1)-category, which Kontsevich models as a linear A A_\infty- or – once rectified/semi-strictified – as a dg-category).

[…] shibboleth […]

I am not a big fan of the shibboleth story. But I strongly agree that it is most curious that after Grothendieck there was this schism in the community, and I am all in favor of discussion that aims to bring together again what belongs together.

a vast generalization of tannaka-krein theorems, about recovering an algebraic stack from its coherent sheaf category.

By the way, there is a nice account of this in

Jacob Lurie, Tannaka duality for geometric stacks (pdf)

It also announces a followup article where the analogue is dicussed for \infty-stacks. But if this followup has already been made available, then I am not aware of it.

my sense of this estrangement between category-theorists and algebraic geometry is based largely on my own personal experiences, my failure for such a long time to understand what algebraic geometers were saying. i don’t really know for sure whether it’s mainly just me who experienced this estrangement, or whether other category-theorists have had a similar experience.

I have the same impression. I was once talking to somebody who made his career in algebraic geometry, and who told me that he just has no chance of understanding synthetic differential geometry. How curious that is: he has been spending his whole academic life inside a topos that models sdg!

What I find most curious also is the widespread habit in algebraic geometry to happily use plenty of general abstract constructions, but at the same time always pick one specific site (or two), without checking much what of the constructions really depend on the choice of that site. I guess that’s also one of the points you have been making here, maybe.

Recently I supervised a master-student who had some fun with showing (here) that a good deal (the higher (𝒪Spec)(\mathcal{O} \dashv Spec)-Isbell duality adjunction) of the story of (derived/higher) algebraic geometry in fact works with the theory of ordinary algebras replaced by any abelian Lawvere theory.

Posted by: Urs Schreiber on December 10, 2010 1:58 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Can you remind me what you mean by “algebroid” please? Is it just an additive category?

Posted by: Mike Shulman on December 4, 2010 3:34 AM | Permalink | Reply to this

Re: Category Theory Puzzle

first i’ll have to find out what you mean by “additive category”. i was under the impression that “additive category” is more ambiguous than “algebroid” is; there seems to be a warning about this at the wikipedia entry for “additive category”, for example. by “algebroid” i mean “v-category” where v is the usual symmetric monoidal category of modules over some unspoken commutative ring. in other words algebroid = “multi-object algebra”, so to speak. (with the more or less usual default assumptions for algebras: unital, associative.)

i have occasion to say “algebroid” somewhat often these days and i find “additive category” annoyingly long to say in its place, for an approximate net gain of ambiguity.

another possibility would be “ringoid”. i know some algebraic geometers who use “ring” for what i would call “algebra”, so i’m not really sure whether there are any good reasons for using “algebroid” instead of “ringoid”. nevertheless i’ve been using “algebroid” recently.

Posted by: james dolan on December 4, 2010 4:27 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Okay, you’re right, “additive category” is also ambiguous. I should have said “Ab-enriched category” for what I meant. Which of course isn’t the same as an R-Mod-enriched category, but for that one could say “R-linear category.”

The problem with “algebroid” for me is that I’ve spent a lot of time listening to people talk about Hopf algebroids, which rather than categories enriched over modules, are groupoids internal to affine schemes (viewed in their dual incarnation as cogroupoids internal to rings). I’m fine with using it to mean “R-linear category” in other contexts, but it helps me to be reminded of the current meaning of ambiguous words at the beginning of any conversation.

Posted by: Mike Shulman on December 4, 2010 6:13 AM | Permalink | Reply to this

Re: Category Theory Puzzle

the comparing times bit started as just a stupid joke when chris rogers and i tried to figure it out and we were amazed by how long it took us to do it.

Posted by: james dolan on December 2, 2010 9:07 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Comment in rot13 and verbose to protect my solution (attempt) from puzzlers. It took me well over 30 minutes and a Wikipedia visit to remind me of the appropriate fundamental properties of bilinearity, and after that it was very fast to wrap it up.

Ol ovyvarnevgl, gur pbzcbfvgvba bs nalguvat jvgu na nqqvgvir vqragvgl vf ntnva na nqqvgvir vqragvgl.
Fvapr gur bowrpg k vf grezvany, gurer vf bayl rire bar fvatyr zbecuvfz sebz na bowrpg l gb gur bowrpg k. Guhf, nal fhpu zbecuvfz unf gb or gur nqqvgvir vqragvgl.
Abj, pbafvqre n zbecuvfz sebz gur bowrpg k gb na bowrpg l. Cerpbzcbfr jvgu gur vqragvgl zbecuvfz ba gur bowrpg k. Guvf vqragvgl zbecuvfz unf gb, ol grezvanyvgl, or gur havdhr zbecuvfz sebz k gb k, naq guhf unf gb or gur nqqvgvir vqragvgl. Guhf, guvf pbzcbfvgvba unf gb or gur vqragvgl. Guhf, gur bayl zbecuvfz gung pna rkvfg sebz gur bowrpg k jvyy or na nqqvgvir vqragvgl. Guhf gur bhgtbvat ubz tebhcf nyfb ner gevivny, cebivat vavgvnyvgl.

Posted by: Mikael Vejdemo-Johansson on December 2, 2010 4:26 AM | Permalink | PGP Sig | Reply to this

Re: Category Theory Puzzle

Nice! — and the use of rot13 is a cute extra touch. Anyone who wants to use this program to decode Mikael’s solution, or encode their own, can do it easily here.

My solution was the same, and so was Jim’s. I wonder if this is, in some sense, the “only” solution.

As for me,

V fcrag yrff guna n zvahgr ernyvmvat gung gurer rkvfgf n zbecuvfz s sebz gur grezvany bowrpg gb k, anzryl gur mreb zbecuvfz. Gur ceboyrz, gura jnf gb fubj gung guvf zbecuvfz vf havdhr.

V fhccbfrq gurer jrer gjb naq gevrq gb cebir gurz rdhny. V jnf dhvgr sehfgengrq, fvapr nyy V xarj jnf gung gurer vf n havdhr zbecuvfz sebz k gb gur grezvany bowrpg. Pbzcbfvat gung jvgu gur zbecuvfzf sebz gur grezvany bowrpg gb k qbrf ab tbbq - ab tbbq va rvgure beqre. Nsgre n srj zvahgrf V tnir hc naq pbagvahrq zl pbairefngvba jvgu Wvz.

Gur arkg zbeavat V gevrq vg ntnva juvyr fvggvat ng gur ohf fgbc. V ernyvmrq gung guvaxvat nobhg gur havdhr zbecuvfz sebz k gb gur grezvany bowrpg jnf hfryrff.

V qrpvqrq gb gel na rnfl fcrpvny pnfr: yrg k or gur grezvany zbecuvfz. Jryy, qhu! Gura gurer’f n havdhr zbecuvfz sebz gur grezvany bowrpg gb k. Vg’f havdhr orpnhfr k vf grezvany. Fb vg zhfg or gur vqragvgl zbecuvfz, naq vg zhfg nyfb or gur mreb zbecuvfz.

Nun: fb gur vqragvgl zbecuvfz sebz gur grezvany bowrpg gb vgfrys vf gur mreb zbecuvfz! Abj, vs V unir nal zbecuvfz s sebz gur grezvany bowrpg gb k, V pna cerpbzcbfr vg jvgu gur vqragvgl zbecuvfz ba k, juvpu vf nyfb gur mreb zbecuvfz, naq pbapyhqr gung s vf gur mreb zbecuvfz. Wbl!

Guvf fgntr gbbx nobhg 5 be 10 zvahgrf. Ubjrire, fyrrcvat ba gur ceboyrz pbhyq unir urycrq.

Posted by: John Baez on December 2, 2010 4:55 AM | Permalink | Reply to this

Re: Category Theory Puzzle

I remember we had to solve this puzzle as an exercise in an intro category theory course. I don’t remember how long it took to solve originally, as everything was equally unobvious at that point, but the solution is cute!

Posted by: Owen Biesel on December 2, 2010 4:30 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Same solution here, but a slightly different take on it:

Zl svefg gubhtug jnf: “pyrneyl gurer vf ng yrnfg bar znc sebz k gb rirel bowrpg, anzryl gur mreb znc”. Gur ceboyrz vf gura gb fubj havdhrarff. Gur sbyybjvat gura pnzr gb zvaq; lbh’yy erpbtavfr vg sebz gur cebbs bs gur fcrpvny nqwbvag shapgbe gurberz:

Yrzzn. Vs k vf na bowrpg bs n pngrtbel P, naq (s_l : k –> l | l va P) vf n pbar jvgu iregrk k bire gur vqragvgl shapgbe ba P, gura k vf vavgvny vss s_k = vq_k.

Cebbs. Gur bayl vs vf pyrne. Sbe gur vs, yrg t : k –> l or nal znc bhg bs k. Abj s_l = t.s_k (nf s vf n pbar) = t (ol nffhzcgvba), drq.

Guvf abj nccyvrf vzzrqvngryl gb bhe fvghngvba; gur snzvyl bs mreb zncf bhg bs k sbez n pbar (ol ovyvarnevgl); naq gur pbzcbarag ng k zhfg or gur vqragvgl (ol grezvanyvgl). Fb jr ner qbar.

Puhaxvat gur nethzrag yvxr guvf vf urycshy, orpnhfr lbh pna abj rnfvyl fpnyr vg hc gb gnpxyr gur ovanel pnfr.

Posted by: Richard Garner on December 2, 2010 6:38 AM | Permalink | Reply to this

Re: Category Theory Puzzle

It took me well over 30 minutes and a Wikipedia visit

Here is an alternative for those with less time. 0 minutes and an nnLab visit: the solution is at zero object (and has been there since Mike added it April last year, but I have expanded on it now).

to remind me of the appropriate fundamental properties of bilinearity

In fact only enrichment in pointed sets (hence the existence of zero morphisms) is needed.

Posted by: Urs Schreiber on December 2, 2010 10:16 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Here is an alternative for those with less time. 0 minutes and an nLab visit: the solution is at zero object (and has been there since Mike added it April last year, but I have expanded on it now).

Only an alternative if you’re not interested in the puzzle aspect — I at least was actually interested in figuring it out myself, and not just reading a proof somewhere.

In fact only enrichment in pointed sets (hence the existence of zero morphisms) is needed.

Sure, though I did end up (re)proving the relevant properties of the zero morphisms from the bilinearity; and thus had use of the glance at wikipedia.

Posted by: Mikael Vejdemo-Johansson on December 2, 2010 3:44 PM | Permalink | PGP Sig | Reply to this

Re: Category Theory Puzzle

Only an alternative if you’re not interested in the puzzle aspect

Sure. I was just kidding. It never works online. Either way.

Posted by: Urs Schreiber on December 2, 2010 4:18 PM | Permalink | Reply to this

Re: Category Theory Puzzle

Sure. I was just kidding. It never works online. Either way.

I don’t know why, exactly, but I often have a really hard time reading your meta-textual moods. I tend to read you far more seriously (and more stand-off-ish) than you expect to be read. It’s not good for our communication. ;-)

Posted by: Mikael Vejdemo-Johansson on December 2, 2010 10:47 PM | Permalink | PGP Sig | Reply to this

Re: Category Theory Puzzle

One’s speed will obviously depend on one’s experiences with category theory.

Here’s a fun fact which is related to the puzzle:

  • Let CC be a category with finite products. Then there is at most one way that CC can be enriched in abelian groups.

In other words, up to natural isomorphism there is at most one way of lifting the ordinary hom to a hom valued in abelian groups

Ab Ob(C)×Ob(C) hom Set\array{ & & Ab \\ & \nearrow & \downarrow \\ Ob(C) \times Ob(C) & \underset{\hom}{\to} & Set }

so that CC becomes enriched in AbAb.

You need to assume finite products for this, and an amusing puzzle is to cook up an example where an ordinary category CC has more than one enrichment.

Posted by: Todd Trimble on December 2, 2010 11:34 AM | Permalink | Reply to this

Re: Category Theory Puzzle

Wnzvr’f “pbhagrerknzcyr” nobir (nybat jvgu nal frg jvgu zber guna bar tebhc fgehpgher) cbvagf gur jnl gb fvzcyr fbyhgvbaf bs guvf rkgraqrq ceboyrz.

Posted by: Peter LeFanu Lumsdaine on December 2, 2010 4:00 PM | Permalink | Reply to this

Re: Category Theory Puzzle

Bu, evtug – V unqa’g gubhtug bs gung! Jung V gubhtug bs vf zhpu zber pbzcyvpngrq naq ybbxrq ng bar-bowrpg pnfrf: gur svryq bs engvbanyf naq vgf rkgrafvba svryq trarengrq ol n fdhner ebbg bs svir unir gur vfbzbecuvp zhygvcyvpngvir zbabvqf (gurve zhygvcyvpngvir tebhcf ner obgu vfbzbecuvp gb gur vagrtref zbq gjb gvzrf n serr noryvna tebhc ba pbhagnoyl znal trarengbef) ohg ner qvssrerag nqqvgviryl.

Posted by: Todd Trimble on December 2, 2010 4:30 PM | Permalink | Reply to this

Re: Category Theory Puzzle

That’s still true if you replace abelian groups with commutative monoids, right?

Posted by: Robin Houston on December 2, 2010 4:11 PM | Permalink | Reply to this

Re: Category Theory Puzzle

Yep! (Good to hear from you again, Robin!)

Posted by: Todd Trimble on December 2, 2010 4:17 PM | Permalink | Reply to this

Re: Category Theory Puzzle

Robin’s appearance made me think of another fun puzzle: in a compact-closed category, is a terminal object always a zero object?

This is true, but I couldn’t prove it without cheating and looking at Robin’s paper. It’s not explicitly a lemma there, but the necessary argument is on the first page.

Posted by: Jamie Vicary on December 3, 2010 10:51 AM | Permalink | Reply to this

Re: Category Theory Puzzle

That’s a nice one! I puzzled over it for a while before coming up with the same solution as Robin. A more explicit proof can of course be extracted from that argument; I wonder whether anyone might come up with the explicit version first?

Posted by: Mike Shulman on December 3, 2010 11:14 PM | Permalink | Reply to this

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