The n-Category Café

When I last checked, the best-developed notion of ‘nerve’ for Batanin’s globular $\omega$-categories was not a simplicial set but one of Joyal’s $\theta$-sets:

• Clemens Berger, A cellular nerve for higher categories.
• Denis-Charles Cisinski, Batanin higher groupoids and homotopy types.
• Eugenia Cheng, Batanin omega-groupoids and the homotopy hypothesis, taped lecture from the Fields Institute Workshop on Higher Categories and their Applications, January 10, 2007.

Maybe Michael or Denis-Charles can step in and tell us what people know today.

When I last checked, the best-developed notion of ‘nerve’ for Batanin’s globular $\omega$-categories was not a simplicial set but one of Joyal’s $\theta$-sets:

Thanks!

One more:

is there a notion of limit in a Batanin $\omega$-category?

Or is there a notion of limit in a Joyal $\theta$-set, alternatively?

That should boil down to asking: is there a good way to say undercategory in Joyal $\omega$-categories / $\theta$-sets?

Surely there must be a simplicial nerve of a $\theta$-set, no?

I feel silly trying to guess answers to these questions given that Michael Batanin, Denis-Charles Cisinski, and other experts read this blog. Someone should wake them up.

I don’t know. I don’t even know about this in the ordinary case that you mention. But I have one general remark concerning this bit:

the completion under filtered colimits of monomorphisms?

I think it is an illusion of low categorical dimension that topos theory is fundamentally related to the notion of monomorphisms and subobjects.

This illusion arises because in the degenerate case of $n= -1$, weak pullback of

$$\array{ {*} \\ \downarrow \\ n Cat }$$

happens to classify monomorphisms. But it is that map ${*} \to n Cat$ which is fundamental, not the notion of monomorphism, and for higher $n$ there is no relation between the former and the latter.

For more on this see section 6.1.6 page 454 of HTT.

I don’t think it’s quite right to say that the ideal completion of Set is “the” category of classes of sets. First of all, in order to make sense of the latter, you need to choose a more expressive set theory than ZFC, and different choices give you different results (even aside from the fact that any given first-order theory has lots of models). For instance, if you use von-Neumann-Bernays-Godel set-class theory, then the category of classes is not cartesian closed, whereas in some stronger set-class theories (such as Grothendieck universes with class := large set) it is. One can axiomatize what is meant by “a” category of classes containing Set (or any other given (pre)topos) as its category of sets, and any (material) set-class theory gives you an example. Now the ideal completion also gives you an example, but I don’t think it can naturally be identified with the one arising from any material set-class theory. In particular, by passing from the class of sets to the category of sets, you lose the ability to distinguish between isomorphic sets, so the ideal completion has to take place at a “structural” level rather than a “material” one. (See here for the meaning of “material” vs “structural”.)

Personally, I think that categories of classes are not really the most natural thing to consider. In most of mathematics, we never really treat about the elements of a proper class (such as the class of sets, or of groups) as elements of a set-like thing, but only as objects of some large category—in particular, we only think about them up to isomorphism. So rather than the category of classes, I think it is more interesting and illuminating to look at the 2-category of large categories. The analogue for an arbitrary topos is the 2-category of stacks on the topos, which has an internal logic and “contains the topos” as an object (via its self-indexing). This is quite close to the ideal completion, since as David quoted the latter can be defined using sheaves. In fact, the ideals living inside sheaves correspond fairly precisely to the “locally small” stacks living inside all stacks.

This suggests that the right analogue of ideal completion for an $n$-topos is its $(n+1)$-category of (locally small) $n$-stacks, where of course $n$ could be $\infty$.