### 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 $(\infty,1)$-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.

I don’t like that there are so many different models in use. Urs lauded the new $(\infty,1)$-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 “$(\infty,1)$-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.

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 $(2,1)$-category). Instead you have to write down some simplicial set that generates it and then take a fibrant replacement.

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.”

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.

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.

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 $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-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 $(\infty,1)$-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!

## 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:

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

identifiyinga concrete representative for an object in $n Cat$ 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 $CartSp$. Many models, none of them really universally suitable for all purposes.

It is precisely the power of category that we do

nothave to worry that there are all these different models for the objects of $Diff$.So why worry much that there are many models for the objects of $(\infty,1)Cat$?