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Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?

Note that this category is closely related to other known models for $(\infty,1)$-categories. For instance, any simplicially enriched category can be regarded as an internal simplicial category with a discrete simplicial set of objects. And any internal simplicial category has a bisimplicial nerve which is a Segal space. One might hope that these functors would be part of Quillen equivalences.

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I'm not aware that anyone has done this. I thought about this idea at one time, but dropped it when I realized that putting a suitable model category structure on simplicial objects in simplicial sets gave everything I wanted (i.e., a cartesian model category). Someone should do this. –  Charles Rezk Dec 23 '12 at 16:36
The second part of my question from earlier: mathoverflow.net/questions/116399/… was precisely motivated by trying to understand and even iterate this suggestion. I assume you want to do this to model (∞,1)-categories within homotopy type theory. Chris Schommer-Pries conjectures it should be possible to model even (∞,n)-categories as strict n-fold categories internal to spaces. Barwick+Kan's n-relative categories let you prove Chris's conjecture on the level of homotopical categories, but so far there are no model structures. –  Jeremy Hahn Dec 23 '12 at 18:07
To answer your question, Barwick and Kan define a relative category to be a category with weak equivalences. The nerve of a relative category is a priori a simplicial object in simplicial sets, but it is not hard to see that it is in fact a strict category object in simplicial sets. They show that every complete Segal space is equivalent to such a nerve. All of this carries over to homotopical versions of nfold categories. –  Jeremy Hahn Dec 23 '12 at 23:18
Surprisingly enough, not everything I do has anything to do with homotopy type theory. I very much doubt this would have any application therein. (-: –  Mike Shulman Dec 24 '12 at 14:26
Nerves of relative categories do seem related to the question, but I don't quite see yet that they answer it. Maybe you could develop your idea further and post it as an answer? –  Mike Shulman Dec 24 '12 at 14:26
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2 Answers

up vote 12 down vote accepted

A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence. In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers.

The above category objects generate a full sub-(relative category) of Rezk's relative category of complete Segal spaces. As I explain below, Barwick and Kan's work proves that the inclusion of this sub-(relative category) induces an equivalence of homotopy theories.

Barwick and Kan construct a nerve functor $N$ from small relative categories to simplicial spaces. The key point is that anything in the image of this nerve is a category object in the above sense.

Their nerve functor $N$ has a left adjoint $K$, but they also consider a second functor $M$ from simplicial spaces to relative categories. The functors $M$ and $N$ are inverse equivalences of homotopy theories in the sense that there is a zigzag of natural weak equivalences $$NMX \rightarrow NKX \leftarrow X$$ for any simplicial space $X$, and a natural weak equivalence $$MNY \rightarrow Y$$ for any relative category $Y$.

If one restricts the domains of $K$ and $M$ to consist only of category objects, the above natural weak equivalences remain intact. Thus the Barwick+Kan homotopy theory of relative categories is equivalent to the theory of category objects in simplicial spaces.

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Ah, I see. So this shows that every complete Segal space is equivalent, in the complete Segal space model structure (and indeed in the Reedy model structure), to the nerve of an internal simplicial category. But without a model structure, does that imply that the latter present the same homotopy theory? E.g. does the full inclusion of categories-with-weak-equivalences induce a DK-equivalence on localizations? –  Mike Shulman Dec 24 '12 at 21:58
I added some more details. Let me know if there is still something left out :) –  Jeremy Hahn Dec 25 '12 at 8:00
Got it; thanks! –  Mike Shulman Dec 31 '12 at 17:48
I've accepted this answer, since it basically answers what I was wondering about, but of course having a model structure would also be nice. It sounds like that is still a research problem for someone, though. –  Mike Shulman Dec 31 '12 at 17:51
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It appears that Geoffroy Horel has solved this problem completely.

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