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I was wondering if there is a model-theoretic way of defining the infinity category of infinity-groupoid objects in a category $C$ (more generally, if $C$ is an infinity category itself, but, right now a 1-category is enough). Is there a model structure on $C^{\Delta^{op}}$ such that those objects which are fibrant and cofibrant correspond to "internal Kan-complexes" in the correct way? So e.g. I want the C-enriched nerve of an actual groupoid object of C to be fibrant and cofibrant in this model structure. If you don't know the answer in general, for now I am mostly interested in the case that $C$ is the 1-category of topological spaces (here I DO NOT want to think of $C$ as being the same thing as infinity-groupoids or simplicial sets, I actually care about the topology).

More generally, if $C$ is an infinity-category associated to a model category $D$, does this correspond to the Reedy-model structure on $D^{\Delta^{op}}$?

EDIT: I should really be asking for a SIMPLICIAL model category structure.

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I don't have an answer which includes the category of topological spaces, but the two remarkable papers "Sheafifiable Homotopy Model Categories I and II" by Tibor Beke together give you a big class of categories with good model structures on their simplicial objects. They are available on his homepage. I think your requirement for internal groupoids is satisfied for these, but better check it out yourself.

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Thanks, I'll check it out. However, I am really interested in an answer which would include C=Top, and C=manifolds. – David Carchedi Mar 23 '10 at 23:52

There is not a model category on simplicial smooth manifolds which gives the data you're looking for, but there is a structure of category of fibrant objects such that nerves of groupoids are fibrant. The fibrations are Kan maps (a la Henriques, but with the additional requirement that the maps on vertices be submersions), and the weak equivalences are maps which induce isomorphisms on all simplicial homotopy groups (again, as Henriques defines them). Nerves of groupoids are fibrant in this sense, hypercovers are trivial fibrations, and morphisms qua principal bibundles are equivalent to morphisms via spans where the source leg is a hypercover.

This category of fibrant objects structure doesn't extend to a model structure on simplicial manifolds because there aren't any positive dimensional cofibrant objects (e.g. hypercovers are trivial fibrations, but given any positive dimensional manifold, you can build a hypercover on it which doesn't admit a section).

If you still want to work in a model category capturing this, the natural thing is to use Yoneda and pass to the local model structure on simplicial presheaves. You can show that this is a fully faithful embedding on the level of homotopy categories.

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Jesse, how does Thm 10.3 of DHI say that this is a faithful embedding? – Thomas Nikolaus Apr 6 '12 at 14:55
Thomas, you're right to ask. I was thinking about it today, and I'm no longer convinced that you can deduce that from their theorem. On the other hand, you can show fairly directly that Yoneda gives a functor of categories of fibrant objects from Kan manifolds to locally fibrant simplicial presheaves (in particular, it preserves fibrations, weak equivalences and path objects). Then, with a little work you can show that any trivial fibration over a representable representable base can be refined to a representable trivial fibration. From this, it follows quickly that it induces a – Jesse Wolfson Apr 6 '12 at 22:21
that it induces a fully faithful embedding at the level of homotopy categories. Finally, using a result of Jardine (Prop. 2.8 of Simplicial Presheaves), we can conclude the statement I made above. – Jesse Wolfson Apr 6 '12 at 22:22
Hm, I still do not see how you show that a trivial fibration over a representable base can be refined to a representable fibration. To be honest I doubt that it is true. Can you maybe give me some more details? – Thomas Nikolaus Apr 6 '12 at 22:39
This isn't so bad. The argument is similar to Lemma 8.6 of Artin and Mazur's Etale Homotopy. The idea is that you can induct up the skeleton so that at the $n^{th}$ stage the $n$-skeleton is representable. The possibility of this refinement comes from the characterization Jardine gives of trivial fibrations in terms of maps of simplicial sheaves such that the maps from the $n$-simplices to the $n^{th}$ matching objects are generalized covers. – Jesse Wolfson Apr 7 '12 at 0:10

Did you check math.AT/0603563 (``integrating $L_\infty$ algebras" by André Henriques)? I am not an expert, but I think he does a construction close to what you are looking for in the category of smooth manifolds.

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