Infinity groupoid objects - MathOverflow

Infinity groupoid objects

2010-03-20T14:10:33Z

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.

Answer by Peter Arndt for Infinity groupoid objects

2010-03-21T15:30:39Z

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.

Answer by yael fregier for Infinity groupoid objects

2011-03-18T11:30:09Z

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.

Answer by Jesse Wolfson for Infinity groupoid objects

2012-04-06T14:42:37Z

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.