The n-Category Café

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

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 $C^\infty$-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 $S^n$. This is kind of what I feel like with $(\infty,1)$-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.

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

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 $n$-sphere $S^n$ is certainly easy to talk about as a topological space, but when $n\gt 1$, 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 $\Pi_\infty$.

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 $n$-dimensional disks together along their boundary $(n-1)$-spheres, while $\infty$-groupoids are built by gluing $n$-morphisms together along their source and target $(n-1)$-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” $n$-disk is an $n$-dimensional sphere, whose $\Pi_\infty$ contains loads and loads of $k$-morphisms for $k\ge n$ (at least as long as $n\gt 1$). On the other hand, a “free-living” $n$-morphism corresponds to the space $K(\mathbb{Z},n)$, which, if you build it out of disks, requires loads and loads of $k$-disks for $k\ge n$. 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.

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 $\omega$-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 $B/Top/B$ of sectioned spaces over $B$: first we take a pullback (limit) over $B$, 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-$\omega$-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 $n$-morphisms” with any class of $(n+1)$-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.