# Models for, and motivation for, (oo,n)-categories for general n

First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category theory for general n? Whatever definition of (oo,n)-categories we use, what are the desired things it should satisfy? What should the main examples be, for general n? I know that bordism categories should be examples. What else? Actually, aside from the cobordism hypothesis and other TQFT-y things I don't really know what the motivations are for (oo,n)-category theory for general n (or at least n bigger than 1), so I hope that people can say some words about that as well. (For n=1, there seems to be a lot of motivation, see for example this or this or this or this or ...)

Second: Presently, what are the models that we have for (oo,n)-categories? Which models have been proven to be equivalent? Of course, there's already a lot about this on the nLab:

(oo,1)-categories

(oo,2)-categories

(oo,n)-categories

I'm mainly just curious about the current status on (oo,n)-categories for general n. Aside from n-fold complete Segal spaces, are there other definitions/"models"?

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Meanwhile the relevant nLab entry has been expanded a fair bit: ncatlab.org/nlab/show/(infinity,n)-category – Urs Schreiber May 18 '12 at 14:05

For me a "model of (∞,n)-categories" is something (e.g., a model category) from which one can extract "the" (∞,1)-category of (∞,n)-categories. One could make this more precise by choosing a preferred definition of (∞,n)-categories and asking for things equivalent to it. Of course it's currently less clear than for, say, models of spaces or spectra that there really is a unique "correct" (∞,1)-category which really is equivalent to everything we hope it's equivalent to. And indeed already for n = 1 there are related but distinct useful notions of category as I write here which manifest themselves in homotopy theory as Segal spaces and complete Segal spaces. But I think most everyone expects that there's a natural notion of (∞,n)-category up to n-categorical equivalence which is the right analogue of n-category (whatever that means).

The easiest examples are of course in the case n = 2, where we have (∞,2)-categorical analogues of the usual examples of 2-categories, for instance the (∞,2)-category of A ring spectra and bimodules, or the (∞,2)-category of (∞,1)-categories, or presentable (∞,1)-categories, or stable (∞,1)-categories, ... For instance if you wanted to understand the relationship between (∞,1)-categories and their stabilizations—(∞,1)-categories form an (∞,2)-category, and stable ones form some kind of subcategory, and you might ask whether there is something like an adjoint to the inclusion—there isn't quite, but maybe there's an adjunction if we view these (∞,2)-categories as objects of some other (∞,3)-category. So (∞,n)-categories do arise naturally in the study of (∞,k)-categories for k < n. These examples are in a sense "algebraic" objects, as opposed to bordism categories, which turn out to be algebraic too, in a sense, but a priori are given by geometric constructions.

As for models for (∞,n)-categories: Besides the iterated complete Segal space model we have Charles Rezk's Θn-spaces, and I think simplicial strict n-categories are also supposed to give the right notion. There's the "complicial sets" model, which seems to me to be more conjectural. I would also like to hear about results about equivalence of these models—as far as I know none have been written down yet, except in the case n = 2.

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I agree mostly, but I would prefer a model for (∞,n)-categories to produce not just the (∞,1)-category of (∞,n)-categories but the (∞,n+1)-category of them. And I would expect simplicial strict n-categories to be too strict. Also, in addition to n-CSS and Θ-spaces, there is a Simpson-Tamsamani approach; I know some people are working on comparing all these, but I don't know of results yet. Finally, any definition of ω-categories, such as Batanin's globular one, can be specialized to (∞,n) by requiring m-cells to be invertible for m>n. – Mike Shulman Dec 27 '09 at 5:39

The nlab page on n-categories includes a list of known definitions and comparisons, which should in particular include all definitions of (∞,n)-categories (since any definition of (∞,n)-category can be specialized to a definition of n-category = (n,n)-category by requiring all k-cells to be trivial for k>n). This list is almost certainly incomplete, but I humbly propose that anyone with knowledge of something it omits should rectify the situation! This sort of question is so common, and the comparisons are so important to the subject, that it would be useful to have a definitive list of what is known.

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Yeah, actually I was planning on prodding anybody who posted a good answer to this question to also contribute to the nLab pages. – Kevin H. Lin Dec 27 '09 at 15:54
If you receive a good answer, repay for that person's effort and turn it into a paragraph on the nLab yourself! Think of the nLab as a place to (among other things) accumulate good MO answers in a stable hyperlinked context. – Urs Schreiber Dec 28 '09 at 14:38