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April 29, 2010

Confessions of a Higher Category Theorist

Posted by Mike Shulman

This is a (long-delayed) continuation of the series begun by Urs and Tom a while ago, about various approaches to higher category theory and how we feel about its current and future directions. Specifically, I want to talk about the new “(,)(\infty,-)-ized” category theory that’s rapidly spreading through mathematics, based on models like quasicategories and complete Segal spaces. As Urs has eulogized at length, it’s really exciting that we now seem to have notions of higher category where one can really develop a theory and start to actually do and apply higher category theory.

On the other hand, I think there are still questions and open directions in higher category theory that aren’t addressed or solved by these models, and thus there are reasons to continue working on alternate models. In particular, I’m not yet ready to declare that quasicategories are The Way to talk about (,1)(\infty,1)-categories; they seem to be quite good for now, but there are things about them that one may still want to improve on. Now I don’t mean to imply that anyone has said that they are The Way (although if anyone out there does feel that way, let’s talk about it!). Rather, what I’d like to do is really have a discussion about what’s good and what’s not as good about these models, and what directions still remain for foundational progress in higher category theory.

In no particular order, here are some of the things I’d like from a theory of higher categories which I don’t feel are supplied (yet) by quasicategories, Segal spaces, and their ilk. But I’d love for someone to tell me that I’m wrong (and why)! Of course, I could be wrong in multiple ways–it could be that the existing models do supply these things and I just don’t realize it, or it could be that I’m wrong to want these things at all.

  1. I don’t like that there are so many different models in use. Urs lauded the new (,1)(\infty,1)-category theory as an “abstract category theory success” which gets away from the kludgy and unnatural notions of model categories to work directly with the deeper structure. But is that really justified, since there is currently not just one “deep/natural” definition of “(,1)(\infty,1)-category”, but rather we have to pass back and forth between several of them using “model categories of homotopy theories”? Ideally, I would like to have a self-contained theory of higher categories which can describe itself directly, without needing to use lots of different models for the same thing.

  2. I like to write down and work with small examples, but this is hard with definitions that are very “flabby.” For instance, it seems nearly impossible to write down explicitly any quasicategory that isn’t actually the nerve of a category (or of something else manageable like a (2,1)(2,1)-category). Instead you have to write down some simplicial set that generates it and then take a fibrant replacement.

  3. I don’t like being tied to one particular shape of cell, particularly simplicial ones. When working with low-dimensional higher categories, like 2-categories and 3-categories, we use pasting diagrams with cells of many different shapes. I would like a notion of higher category that has an underlying sort of “higher computad” where we can talk about arbitrary pasting diagrams and “higher surface diagrams.”

  4. In a notion of higher category like a quasicategory, the higher cells are “doing double duty:” on the one hand they are the morphisms of whatever dimension, but they are also telling us about how the lower-dimensional morphisms compose. I find this conflation of cells with composition operations suboptimal: I would rather be able to give you the cells first, and then tell you how to compose them. Possibly this is related to the “algebraicity” in the strongest version of the homotopy hypothesis that Tom was talking about.

  5. I would like a higher category theory which can be generalized easily to an enriched theory. This seems to be more accessible with some definitions than others. For instance, quasicategories seem quite far removed from any sort of enrichment, while with Segal categories it is usually somewhat clearer what to do, at least in the case of cartesian enrichment. For non-cartesian enrichment, the situation is a good deal murkier.

  6. Finally, I don’t like having to use lots of machinery that I don’t understand. I love machinery, but only when it’s doing something for me that I could understand on its own (though perhaps not prove rigorously), and moreover I know in principle how the machinery is doing what it does. It does seem unlikely that higher category theory will ever be accessible without a fair amount of machinery, but I think it should be possible to make that machinery more comprehensible.

This last point bears some more explanation, as perhaps it says more about me than it does about higher categories. Right now all these theories make me a bit uncomfortable, because I have this probably-unfounded feeling that they depend on huge and complex machines that ordinary mortals can never really understand, only make use of. I think part of this comes from the fact that whenever anyone gives a talk about (,1)(\infty,1)-categories, they always give an “intuitive” description but never want to get pinned down on any details. So I have an overload of intuition by now, but that’s not the same as understanding how something works. (I don’t necessarily mean “details,” but some details are necessarily a part of it.)

Now of course this might be (perhaps even “probably is”) just because I haven’t yet put in the effort to understand what’s going on. What I’d really like is to be as comfortable with (,1)(\infty,1)-categories as I am with 1-categories. For example, if I’m reading a paper on category theory and someone defines some structure that I haven’t seen before, I can usually instantly understand what it’s supposed to do, and why their coherence axioms make sense. Or if someone states a lemma about the behavior of some limits or colimits, even if I’ve never seen it before, I can usually guess how the proof is going to go, based on experience manipulating similar things. But when I read (,1)(\infty,1)-category theory, I am constantly getting run over by various maps being this or that sort of fibration, various kinds of products and joins and slices, manipulations of limits and colimits that I don’t have confidence behave in the way I would hope them to, and so on.

Now it occured to me that some other people might conceivably be in the same boat as well, and so I thought it might be nice to run a series of blog posts aimed at getting over that hurdle: reaching the same comfort level with (,1)(\infty,1)-categories that many of us have with 1-categories. To help start us off, in the Chicago proseminar this quarter we’ve having a sequence of talks with a similar goal, and we’ll try to post notes here from some or all of them. I should say that this is not aimed at working through any particular paper or book. Instead we want to understand things relative to our own points of view, hopes, and expectations, and maybe even end up synthesizing multiple threads.

I started out our seminar a few weeks ago by talking about Thomas Nikolaus’ recent theory of algebraically fibrant objects, which we’ve already discussed here. Then Emily Riehl talked about the adjunction between simplicial sets and simplicial categories in terms of Dugger and Spivak’s “necklaces;” notes from her talk should be appearing here shortly. Here’s to a lively and informative ensuing discussion!

Posted at April 29, 2010 2:53 AM UTC

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Re: Confessions of a Higher Category Theorist

Hi Mike,

thanks for continuing this discussion. Here is a very unimportant comment on one of your points. You write:

I don’t like that there are so many different models in use.

Similar statements have been made by others here before. While I do of course see what you mean, I also do find it a bit curious that otherwise die-hard category theorists do indeed care so much about identifiying a concrete representative for an object in nCatn Cat and are unhappy with the fact that these objects are naturally available only up to equivalent incarnations. Don’t you feel a bit evil about it?

Clearly in practice it is useful to pick a representative that supports most of the operations that one wants to do on it. Just as picking a skeleton of a category is in practice useful only to the extent the things I want to do with the category don’t always force me away from that choice of skeleton. But on abstract grounds, I couldn’t care less about these choices.

And I think we all know lots of examples of categories for whose objects we use different models all the time, and where we are forced to switch between different models all the time, and where there does not seem to be an all-purpose canonial model. Let’s see, take something mundane as the category of manifolds. We are so used to thinking of its objects only up to equivalence, that we have already forgotten how they are really defined by rather explicit and to some extent awkward models. For some puroposes I think of them as representations of pseudogroups, for others as collections of charts, for others as locally representable sheaves on CartSpCartSp. Many models, none of them really universally suitable for all purposes.

It is precisely the power of category that we do not have to worry that there are all these different models for the objects of DiffDiff.

So why worry much that there are many models for the objects of (,1)Cat(\infty,1)Cat?

Posted by: Urs Schreiber on April 29, 2010 8:30 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

So why worry much that there are many models for the objects of (,1)Cat(\infty,1)Cat?

I was thinking about a reply to this, but then the Cafe went down for a while and I forgot about it. Let me try to construct something.

You are of course right that there are lots of situations in which we have multiple models of something, and that that is often part of the power of a notion. I think maybe what I don’t like is a combination of two things:

  1. A perception that no one model is sufficient by itself even for developing the basic theory, and
  2. A perception that the translation between models is difficult.

Being able to move between models is a great thing; being forced to move between models is less so, especially when learning the basic theory. Imagine if you had to learn about charts and representations of pseudogroups and locally C C^\infty-ringed spaces and locally representable sheaves all in your first course on differential geometry, and you needed all of those notions even in order define the manifold S nS^n. This is kind of what I feel like with (,1)(\infty,1)-categories: even in order to define the quasicategory of \infty-groupoids, we have to first define it as a simplicial category and then take its homotopy coherent nerve.

Now I feel somewhat less scared of the homotopy coherent nerve after Emily’s nice post explaining it, and I hope I’ll feel even less scared of it after she talks about the straightening construction in a couple of weeks. So while I would still rather not be forced to use it, perhaps my unease will evaporate with greater familiarity.

Posted by: Mike Shulman on May 14, 2010 4:38 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I’d disagree with the idea that passing between models is necessary for developing the basic theory.
One example under discussion is the (infty,1)-category of spaces, which in my book is described as the quasicategory S given by the coherent nerve of the simplicial category of Kan complexes. But this is only one of many ways to describe S. For example:

a) S is the quasicategory freely generated under small colimits by one generator. (Universal property for mapping out.)
b) For any domain category C, homotopy classes of functors C -> S are in bijection with homotopy equivalence classes of left fibrations over C. (Universal property for mapping in.)

Either of these properties characterizes S up to equivalence. Of course, you at some point need to show that such an S exists. For this, it is convenient to describe S as a homotopy coherent nerve. But this does not really require developing a theory of simplicial categories or even defining the homotopy coherent nerve in general: you could just explicitly write down what the construction gives in this case (vertices = Kan complexes, edges = maps between Kan complexes, etcetera). Or give some other construction.

By analogy: in the theory of groups, the group
Z plays an important role. It is the free group on one generator. A fancy way to say this is that the forgetful functor G: Groups -> Sets has a left adjoint F, and that Z=F(*). But the fact that Z can be described in this way isn’t really an indication that the theory of groups is not self-contained, and that you need the technology of categories and adjoint functors to understand even the simple examples.

Posted by: Jacob Lurie on May 15, 2010 12:08 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

That’s an interesting point. How far could you get if you just took the existence of a quasicategory SS with one or the other of those properties as an axiom?

Is there nowhere else in the basic theory that you need to use simplicial categories? I feel like a lot of the time the quasicategories we are interested in are the localizations of model categories, and those seem also to be most naturally constructed as homotopy coherent nerves; the situation isn’t limited to SS. I guess some of the time, at least, those quasicategories could also be constructed directly quasicategorically (such as for (,1)(\infty,1)-topoi).

Posted by: Mike Shulman on May 15, 2010 2:12 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I wrote:

…a lot of the time the quasicategories we are interested in are the localizations of model categories, and those seem also to be most naturally constructed as homotopy coherent nerves…

After writing this, I realized that it is probably an unfair objection. It seems highly likely that whatever notion of (,1)(\infty,1)-categories one uses, an arbitrarily chosen model category will require a fair amount of massaging in order to get into that framework. The only (,1)(\infty,1)-categories we could reasonably expect to construct directly are either ones given by some explicit presentation, or by starting from the “basic” Gpd\infty Gpd and applying various (,1)(\infty,1)-categorical constructions.

So if, for instance, we can write down axioms describing the necessary properties of the (,1)(\infty,1)-category Gpd\infty Gpd in the same way that we can write down axioms describing the necessary properties of the 1-category of sets (like ETCS), then we ought to be satisfied with that. But is it enough to say that it’s the free cocompletion of a point? For instance, the category of sets is always the free cocompletion of a point, but that’s not enough to tell you that it’s an elementary topos; even in predicative set theory the category of sets is the free cocompletion of a point.

Posted by: Mike Shulman on May 15, 2010 5:25 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I’m not really sure I understand your question:
are you asking how many of the equivalent definitions of S will remain equivalent in weaker set theories? If so, then I don’t have much to say other than it would surely depend on the weakening (for example, if you throw out the axiom of choice then you probably don’t want to use the homotopy coherent nerve of Kan complexes, because a trivial Kan fibration might not admit a homotopy inverse).

But this seems to be a rather different question than the one we started with: namely, whether it is possible to give a self-contained development of the theory of quasi-categories (or some other version of the theory of (infty,1)-categories).

The answer to this is surely “yes”. To the extent that such an development is not carried out in existing sources, I think it owes more the the paucity of the literature than to any mathematical difficulties.

Posted by: Jacob Lurie on May 19, 2010 9:04 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

are you asking how many of the equivalent definitions of S will remain equivalent in weaker set theories?

No, that’s not really what I had in mind. Urs got closer; I didn’t really mean “axiom” in the sense of an axiomatic foundational theory. Instead, I meant something more like “how much of the theory can you develop without constructing SS if you just assume that the free cocompletion of a point exists and call it SS?”

This is kind of similar to what I was driving at here with the suggestion that the homotopy 2-category should have a Yoneda structure, and that much of the theory of (,1)(\infty,1)-categories should be expressible purely in terms of that Yoneda structure. Having a Yoneda structure is more or less the same thing as having free cocompletions, relative to some reasonable notion of (weighted) colimit.

whether it is possible to give a self-contained development of the theory of quasi-categories (or some other version of the theory of (,1)(\infty,1)-categories).

The answer to this is surely “yes.” To the extent that such an development is not carried out in existing sources, I think it owes more the the paucity of the literature than to any mathematical difficulties.

That’s good to hear. I hope that the literature eventually fills in the gaps like this.

Posted by: Mike Shulman on May 22, 2010 2:20 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

But this seems to be a rather different question than the one we started with

I think Mike is still driving at one aspect of the original question: it is a bit unpleasant, in some sense, to define the (,1)(\infty,1)-category of \infty-groupoids or that of (,1)(\infty,1)-categories as the hc-nerve of a sSetsSet-subcategory of sSetsSet, instead of directly without passing through simplicial categories.

In a previous thread we had tried to discuss this before. There we tried to see if we couldn’t more directly define a quasi-category

Grpd:[n]LFib(Δ n) 0. \infty Grpd : [n] \mapsto LFib(\Delta^n)_0 \,.

Posted by: Urs Schreiber on May 19, 2010 9:46 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

As to the other question: whether or not it is easy to realize the underlying (infty,1)-categories of model categories. I would say the answer is generally “yes” in cases of interest, for exactly the reasons you describe: model categories that come up most often are built from some primitive examples (like simplicial sets) by categorical constructions (which are usually easier to carry out in the setting of quasi-categories than in the setting of model categories).

The first “counterexample” that comes to my mind would be the category of commutative differential graded algebras over a field of positive characteristic. I have no idea how to give a satisfying intrinsic description of the associated quasi-category. But I also have no idea what to do with the model category. (I have never seen it come up except in the context of “warning: things will go wrong in positive characteristic”, but that may be my own ignorance).

Posted by: Jacob Lurie on May 19, 2010 9:51 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

What about the Strom model structure on topological spaces, where the weak equivalences are the (not weak) homotopy equivalences? Or the similar model structure on chain complexes (due to Cole, I believe) where the weak equivalences are the chain homotopy equivalences?

Posted by: Mike Shulman on May 22, 2010 12:49 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

It seems to me that addressing issue 2) is a tall order: a formalism which would allow you to compute easily with small examples should give you a recipe for calculating unstable homotopy groups of spheres. (Of course, this depends on exactly what you mean by “small.”)

Posted by: Jacob Lurie on April 29, 2010 2:34 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Here’s a silly question: why are homotopy groups hard, really? Is it because K(Z,n)K(Z,n)’s tend to largeness? Is it not knowing when you have enough generators untill you have too many?

Posted by: some guy on the street on April 29, 2010 3:43 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Interesting point! I think it really does depend on what you mean by “small.” I might argue that just because a space is small and easily describable, it doesn’t follow that its fundamental \infty-groupoid should be considered small or easy. The nn-sphere S nS^n is certainly easy to talk about as a topological space, but when n>1n\gt 1, its fundamental \infty-groupoid has lots and lots of higher morphisms in it. So I don’t think the complicatedness of the homotopy groups of spheres necessarily means that we shouldn’t expect to be able to compute with “small” \infty-categories; the complication is produced by the functor Π \Pi_\infty.

Posted by: Mike Shulman on April 29, 2010 5:09 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Is this really the case? At least part of the complication surely comes from the fact that colimits of ω\omega-groupoids (indeed, of anything weak-higher-category-like) can blow up quite unwieldily, and hence ω\omega-groupoids with small presentations needn’t end up simple?

For instance, take the ω\omega-groupoid that (naïvely) “looks like the 2-sphere”, made by freely gluing two 0-cells, two 1-cells, and two 2-cells (“the two hemispheres of the sphere”) in the appropriate ways. This is a pretty small presentation — a computad with just 6 basic cells. But won’t it blow up to significant complexity in higher dimensions, comparable to the higher homotopy groups of the 2-sphere? Or am I missing something?

(Unfortunately I’m rather illiterate on the sides of this question tending more towards “traditional” homotopy theory, so I may well be missing some well-known things here, and/or falling prey to the mis-intuition Mike describes below.)

Posted by: Peter LeFanu Lumsdaine on April 30, 2010 4:13 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Thanks, I was getting a bit overenthusiastic. I think we are both half right.

It’s true that colimits and presentations of higher categories can introduce a plethora of nontrivial higher coherence cells. But I believe the computad you describe presents a very simple ω\omega-category, with no interesting kk-morphisms for k>2k\gt 2. In order to get the 2-sphere S 2S^2 as an ω\omega-groupoid (which is what you said), you have to then invert everything, i.e. reflect into ω\omega-groupoids, and that’s when all the higher homotopy groups pop up.

So it is sort of as I was saying, if you change what I said about kk-morphisms in ω\omega-groupoids to kk-morphisms in ω\omega-categories, and also forget everything I said about K(,n)K(\mathbb{Z},n). (That was rubbish, since K(,n)K(\mathbb{Z},n) is also an ω\omega-groupoid—I was thinking of the free ω\omega-category on a kk-morphism, which I still think is fairly simple, but the free ω\omega-groupoid on a kk-morphism is S kS^k, not K(,n)K(\mathbb{Z},n), and thus not very simple). Free-living kk-morphisms in ω\omega-categories are fairly simple, but when we regard an ω\omega-category as a presentation of an ω\omega-groupoid, then all sorts of complicated things arise that aren’t visible in the original ω\omega-category. (Kind of like how any homotopy type can be modeled by the nerve of a 1-category—you get all the nontrivial higher cells when you formally invert lower-dimensional ones.) So another way to phrase my original response to Jacob’s comment would be that “the ω\omega-groupoid reflection of a small and manageable ω\omega-category need no longer be particularly small or manageable.”

Posted by: Mike Shulman on April 30, 2010 2:40 PM | Permalink | PGP Sig | Reply to this

Re: Confessions of a Higher Category Theorist

I’ve been thinking about this question some more, and I think there’s a sense in which this “mismatch” is where the real meat of the homotopy hypothesis lies. Moreover, one could perhaps argue that it’s precisely this point which the answer “define an \infty-groupoid to be a Kan complex” glosses over.

The homotopy hypothesis is about a connection between (say) CW complexes and \infty-groupoids. Now CW complexes are built by gluing nn-dimensional disks together along their boundary (n1)(n-1)-spheres, while \infty-groupoids are built by gluing nn-morphisms together along their source and target (n1)(n-1)-morphisms. Now disks look a lot like morphisms, which leads people to use the same word “cells” for both of them, but they’re really quite different beasts. A “free-living” nn-disk is an nn-dimensional sphere, whose Π \Pi_\infty contains loads and loads of kk-morphisms for knk\ge n (at least as long as n>1n\gt 1). On the other hand, a “free-living” nn-morphism corresponds to the space K(,n)K(\mathbb{Z},n), which, if you build it out of disks, requires loads and loads of kk-disks for knk\ge n. But it’s easy to forget about this because a one-dimensional disk is the same as a one-dimensional morphism.

The homotopy hypothesis says that nevertheless, you get equivalent notions by gluing together disks and by gluing together morphisms. Put that way, it starts to sound like the “co-cellular” picture in classical homotopy theory where a space is decomposed not as a cell complex, but as a Postnikov tower, although in that case we are “co-gluing” things together rather than gluing them. And in fact, we know that the categorical representation of spaces is well-adapted to characterization in terms of k-invariants, e.g. the characterization of 2-groups using crossed modules and group cohomology classes.

Posted by: Mike Shulman on April 29, 2010 9:26 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Point 4) is an interesting one, and is something I’ve been thinking about lately. It seems that asking composition to be given separately from the morphisms of the next higher dimension is essentially asking for a globular model. If there is a specified composite of any pasting diagram of n-cells, then (n+1)-cells of arbitrary shape can be reduced to globular cells by replacing lower-dimensional pasting diagrams by their chosen composites.

It feels more natural to me to allow these higher-dimensional cells of arbitrary shape, and to let composition be a “virtual operation”. To take the classic example, consider the monoidal category of vector spaces as a bicategory. The composition (tensor product) can be given algebraically, by the standard construction, but this feels uncategorical and evil. Better is to expand the notion of 2-dimensional morphisms to include domains of lists of vector spaces, with morphisms given by multilinear homomorphisms. Then the composition is given (only up to isomorphism) by the usual universal property. I suppose this would give an opetopic bicategory.

Recently I’ve been trying to understand Makkai’s work along these lines. Anafunctors feel like the right middle ground between the algebraic and geometric definitions, and his concept of FOLDS equivalence seems to be a powerful general notion of higher-categorical equivalence.

I do agree with much of your other points. In particular, I still hope for a more canonical feeling path to (∞,1)-categories.

Posted by: Patrick Schultz on April 29, 2010 7:10 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I definitely agree that composition is often an anafunctor, and I have some liking for the opetopic approach. Certainly some bicategories, such as the one you describe, are very naturally defined opetopically, just as some monoidal categories are very naturally defined as representable multicategories. In my comments I was thinking primarily about the very prevalent simplicial models, and the opetopic approach is certainly more natural than simplicial ones in this regard. Simplicially we only have 2-morphisms going from a composite of two 1-morphisms to a single 1-morphism, and we only have 3-morphisms going from a particular sort of composite of two 2-morphisms to a similar composite of two 2-morphisms, and so on, whereas the “unbiased, many-to-one” nature of opetopes, in which we allow arbitrary composites in the source and only single cells in the target, is certainly a more natural way to describe these structures. (I believe the original Baez-Dolan definition of opetopes was specifically motivated to improve on Street’s simplicial ω\omega-nerves in precisely this way.)

However, I would argue that not all “anabicategories” are naturally opetopic ones. In other words, it’s not always the case that composites are naturally defined by first defining the maps out of them. Certainly some monoidal categories, like that of vector spaces and tensor product, are naturally defined as representable multicategories. However, some other monoidal categories, like cartesian monoidal ones, are more naturally defined as co-representable co-multicategories, since the product is determined by maps into it. Still other monoidal categories are defined in neither of these ways: their monoidal product could be defined by a combination of limit and colimit constructions, so that it isn’t defined purely in terms of maps into it or maps out of it. For instance, this is the case for the “fiberwise smash product” on an over-under-category like the category B/Top/BB/Top/B of sectioned spaces over BB: first we take a pullback (limit) over BB, then we take a pushout (colimit) to quotient out the sections. They are still “ana-monoidal categories” in that their monoidal product is only defined up to canonical isomorphism, but they don’t occur “in nature” as a representable multicategory. We can of course make them into one, but only in the way that any monoidal category can be made into a multicategory.

So while I like the idea of making composition an ana-operation, it doesn’t feel to me as though opetopic things are the most natural way to describe all such ana-ω\omega-categories. In other words, it’s good to have many different “reasons” why “the” composite of a bunch of cells should be a particular thing or another particular thing, without forcing ourselves to make a specific choice. But I would rather not be forced to always identify the “ways to compose a bunch of nn-morphisms” with any class of (n+1)(n+1)-morphisms.

This seems to be leading in the direction of Definition L’ from Tom’s survey, a definition which I’ve always thought has received surprisingly little attention.

Posted by: Mike Shulman on April 29, 2010 10:11 PM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I completely agree about the restriction of input types in simplicial models. This is what feels so non-canonical about these models.

You make a very good point about defining higher categories by representability. The asymmetry in the direction of arrows is another non-canonical choice a “right” definition would probably lack.

Definition L’ is very interesting. Skimming over the definition, two things strike me: one is that a category is again a particular type of multicategory, but using contractibility rather than representability. I wonder if there is any relation between these ideas. The other thing that jumps out is the formal similarity with anafunctors. I want to say L’ defines a category as an “ana-algebra” for the free ω-category monad on globular sets. This would be very pleasing if it could be made precise.

Posted by: Patrick Schultz on April 30, 2010 1:12 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I’m very intrigued to see this cropping up — I’ve also been coming round more and more to liking definition L’ recently!

One possible approximation to the “ana-algebra” statement: TT (the free ω\omega-category monad) lifts to a pseudo-monad on the bicategory wk\mathcal{E}_{wk} of “globular sets sets; spans with contractible left leg; maps of spans”. The colax algebras for this monad are then equivalent to the L’-categories, if I’m not mistaken; and this bicategory can of course very reasonably be called the bicategory of “ana-maps” of globular sets.

(Of course, this should probably all be done in a slightly more refined way, to talk about “maps with contraction” rather than “contractible maps”! It also relies on very few of the the specifics; I think it doesn’t even rely on the cartesianness of TT, without which several pseudo- things would become lax, but I think the equivalence would still go through.)

Despite the main contrast Mike draws between the simplicial and globular approaches — that in simplicial approaches, the “witnesses for compositions” are identified with a certain class of the higher-cells — L’-categories seem very similar in flavour to me to Verity’s simplicial model, “weak complicial sets”. In each case, the compositional structure is carried individually by each ω\omega-category (not parametrised once and for all as in the operadic approaches), but is still (unlike in other simplicial approaches) divided into two parts: “a composite of composites is a composite” (the monoid/multicategory structure of an L’-category; the thinness extensions in a weak complicial set), and “any pasting diagram has a composite” (the contractibility in an L’-category; the horn-filling extensions in a complicial set).

Posted by: Peter LeFanu Lumsdaine on April 30, 2010 3:34 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

TT (the free ω\omega-category monad) lifts to a pseudo-monad on the bicategory wk\mathcal{E}_{wk} of “globular sets sets; spans with contractible left leg; maps of spans”. The colax algebras for this monad are then equivalent to the L’-categories, if I’m not mistaken; and this bicategory can of course very reasonably be called the bicategory of “ana-maps” of globular sets.

That is definitely a nice way of thinking about it; that hadn’t occurred to me before. Although I think you want lax algebras, not colax ones.

My first question after thinking about it from that perspective was: why lax algebras and not pseudo ones? But I think the answer is that the 2-cells in that bicategory are not “transformations between anafunctors,” but more like “inclusions of anafunctors.” So being a lax algebra in that bicategory really is like being a pseudoalgebra in a fairly strict way, since if one anafunctor is included in another then they are equivalent.

Posted by: Mike Shulman on April 30, 2010 4:31 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Although I think you want lax algebras, not colax ones.

Oops! I can never remember which way round lax and colax go… I need to find a good mnemonic :-P

So being a lax algebra in that bicategory really is like being a pseudoalgebra in a fairly strict way, since if one anafunctor is included in another then they are equivalent.

Ah, yes — that’s a very nice point! I’d just noted the equivalence, and never thought to ask “why lax?”…

Posted by: Peter LeFanu Lumsdaine on April 30, 2010 5:13 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

I can never remember which way round lax and colax go… I need to find a good mnemonic.

I always start from this, which is easy to remember:

  • Lax monoidal functors preserve monoids. Colax monoidal functors preserve comonoids.

and move up to this:

  • A lax monoidal functor is a lax TT-morphism, where TT is the 2-monad on CatCat whose algebras are monoidal categories.

and that tells you most of what you want to know directly. For instance, it tells you which of lax and oplax natural transformations should be which, if you let TT be the 2-monad on D ob(C)D^{ob(C)} whose algebras are 2-functors CDC\to D. (Although one should be aware that there are some people who use “lax” and “oplax” reversed in that particular case.) For the case at hand, you need one more level-up:

  • A lax TT-algebra structure on an object AA of a 2-category 𝒦\mathcal{K} is a lax MM-morphism TA,AT \to \langle A,A\rangle, where MM is the 2-monad on endo-2-functors whose algebras are 2-monads, and A,A\langle A,A\rangle is the codensity monad of AA, i.e. the right Kan extension of 1A𝒦1 \overset{A}{\to} \mathcal{K} along itself.

Although that last one is complicated enough to work out explicitly that when I need to know what a lax algebra is, I usually just go look it up. And you just motivated me to put it on the nlab, so now we can all look it up in the same place.

Posted by: Mike Shulman on April 30, 2010 2:52 PM | Permalink | PGP Sig | Reply to this

Re: Confessions of a Higher Category Theorist

Of course, this should probably all be done in a slightly more refined way, to talk about “maps with contraction” rather than “contractible maps”!

Why? I like contractible maps much better. They are also much more “ana”; it seems to me like a “map with contraction” would be like an anafunctor with a given choice of a specification for each object!

I think it doesn’t even rely on the cartesianness of T, without which several pseudo-things would become lax, but I think the equivalence would still go through.

That’s right. If TT is any pullback-preserving monad on a category CC with pullbacks, then it lifts to a strong monad on the proarrow equipment Span(C)Span(C). This induces a “monad” on the bicategory Span(C)Span(C) whose functor part is strong, but whose multiplication and unit are only oplax. (I put “monad” in quotes because there is no 2- or 3-category containing oplax transformations, so it’s not immediately obvious what sort of “monad” this is.) If TT doesn’t preserve pullbacks, then the monad it induces on Span(C)Span(C) is also only oplax. In either case, you can still define what you mean by “TT-multicategories;” this was originally done by Burroni, and in our paper Geoff and I put it in the general context of equipments.

the compositional structure is… divided into two parts: “a composite of composites is a composite” (the monoid/multicategory structure of an L’-category; the thinness extensions in a weak complicial set), and “any pasting diagram has a composite” (the contractibility in an L’-category; the horn-filling extensions in a complicial set).

This analogy doesn’t seem very strong to me. It’s true that both have a notion of “any pasting diagram has a composite,” but of course that condition is also present in pretty much any notion of higher category, both algebraic and nonalgebraic. And the notion of “pasting composite” is very different in different cases; in particular for L’ the pasting composites are globular diagrams, while for complicial sets the pasting composites are thin inner horns, which seem about as different-looking as you can get.

And while it’s true that the thinness structure in a WCS exhibits the composites, it also tells you which simplices are supposed to be regarded as “equivalences,” and thereby which diagrams you should expect to be able to compose in the first place. (This is another big difference with L’ or any other definition where “being an equivalence” is a definable property, rather than a specified structure.) It seems to me that the thinness of composites of thin things is more about “the composite of equivalences is an equivalence.” I see “the composite of composites is a composite,” aka generalized associativity, as living more in the existence of fillers for higher-dimensional horns. But that’s all subjective, of course.

Posted by: Mike Shulman on April 30, 2010 3:55 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

On the L’-category versus weak complicial sets question, I think where we’re diverging is that as I see it, the “thin simplices” of a WCS are not exactly the ones “supposed to be regarded as equivalences”. Indeed, they’re not typically all the equivalences in the WCS: they’re a specified class of equivalences chosen to witness composition. You can have e.g. a thin 2-simplex exhibiting h as a composite of f•g, and an “equivalence” in a defined sense between h and h’, but no thin 2-simplex exhibiting h’ as a composite of f•g. (And you can have a very closely analogous situation in an L’-category…)

So I’d think of the thin simplices as a specified class of “witnesses for composition”, residing as a subset of the defined class of “equivalences”; and hence I’d put WCS’s on the other side of the “another big distinction” that you draw!

Posted by: Peter LeFanu Lumsdaine on April 30, 2010 5:04 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Indeed, to get a bit closer, suppose we had something like an opetopic analogue of L’-category. (At least up to 2-d I think it’s clear roughly what that should mean.) Then, like in a WCS, every witness for a composition of n-cells would be realisable as some (n+1)-cell equivalence; though there’s still the big difference remaining on this front that in a WCS, the map {witnesses} \rightarrow {equivalences} is always monic, whereas in an L’-category, we could have multiple witnesses realised as the same equivalence. Could one develop a version of WCS’s in which the map TX nX nTX_n \rightarrow X_n wasn’t necessarily monic, I wonder?

Posted by: Peter LeFanu Lumsdaine on April 30, 2010 5:24 AM | Permalink | Reply to this

Re: Confessions of a Higher Category Theorist

Ah, I see what you were saying. You really did mean “weak complicial sets,” not “Street’s notion of weak ω\omega-categories,” which is what people sometimes mean when they say “weak complicial sets.” (-: You’re right, that is closely analogous, and you can have a similar situation in an L’-category. And unless I’m mistaken, L’-categories include strict ω\omega-categories and strict maps between them as a full subcategory (those where the structure anamap is a strict map), just as WCS do (as strict complicial sets).

I had been thinking that one would need to define “weak morphisms” of L’-categories in order to get a good notion of them, which would have all the attendant complications that defining weak morphisms between Batanin-style operadic ω\omega-categories does. But the analogy with WCS suggests that instead one should restrict to those L’-categories such that the composition anamap is a “saturated” anafunctor in some sense, and then the ordinary “strict” maps of L’-categories might be sufficient.

I wonder how hard it would be to define an L’-category of L’-categories. As you probably know, part of the problem with trying to define the ω\omega-category of ω\omega-categories in Batanin’s world is that you need to cook up a new operad: the collection of PP-algebras for some globular operad PP isn’t itself a PP-algebra, since composing kk-transfors requires composing nn-cells in the codomain for many different values of nn. But L’-categories seem like they might sidestep that problem.

Posted by: Mike Shulman on April 30, 2010 3:09 PM | Permalink | PGP Sig | Reply to this

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