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July 29, 2009

Question on Models for (∞,1)-Functor Categories

Posted by Urs Schreiber

Here is a question on models for (,1)(\infty,1)-categories of (,1)(\infty,1)-functors.

Let CC be a simplicial model category (let’s say it’s also combinatorial) and C C^\circ the (,1)(\infty,1)-category presented by it.

Let RR be any ordinary locally small category, regarded as a simplicially enriched category.

Then on the SSet enriched functor category [R,C][R,C] there is the global injective model structure on functors.

Question 1: Under which conditions is [R,C] inj[R,C]_{inj} a simplicial model category? Under which conditions does it present the (,1)(\infty,1)-categoryof (,1)(\infty,1)-functors Func (,1)(R,C )Func_{(\infty,1)}(R, C^\circ)?

In HTT this is answered for the case that C=SSetC = SSet: then [R,C] inj=[R,SSet] inj[R,C]_{inj} = [R,SSet]_{inj} is the global injective model structure on simplicial presheaves that does indeed present the (,1)(\infty,1)-category of (,1)(\infty,1)-presheaves on RR for all RR.

What is the statement for more general CC?

Now consider the special case that RR is a Reedy category. Then by a theorem by Vigleik Angeltveit the Reedy model structure on the simplicially enriched functor category [R,C][R,C] makes [R,C] Reedy[R,C]_{Reedy} into a simplicial model category.

Question 2: Under which conditions does [R,C] Reedy[R,C]_{Reedy} present the (,1)(\infty,1)-category of (,1)(\infty,1)-functors Func (,1)(R,C )Func_{(\infty,1)}(R, C^\circ)?

Posted at July 29, 2009 1:24 PM UTC

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Re: Question on Models for (∞,1)-Functor Categories

Before his untimely death, Thomason was working on a new definition of model categories. The main point of this definition is supposed to be that any category of functors from a small category into a Thomason model category is again a Thomason model category. I am guessing that the corresponding statement should be true for the right notion of simplicial Thomason model category as well. As far as I know, the only place to read about some of this is Weibel: Homotopy ends and Thomason model categories. This article is a bit unsatisfactory, in that it does not prove the above statements in full generality. However, work is currently being done on the subject, by Xuan Yang and possibly by others as well.

Anyway, it might be that the right context for asking questions about model structures on functor categories is the setting proposed by Thomason. I don’t know if it is easy to make precise the idea of “the (,1)(\infty, 1)-category presented by a simplicial Thomason model category”. If one can do this, it is conceivable that there are no conditions at all needed for the questions in the post.

Posted by: Andreas Holmstrom on August 9, 2009 7:11 PM | Permalink | Reply to this

Re: Question on Models for (∞,1)-Functor Categories

Thanks for this information about Thomason model categories. I hadn’t been aware of that notion. I’ll have a look.

Posted by: Urs Schreiber on August 10, 2009 7:44 PM | Permalink | Reply to this

Re: Question on Models for (∞,1)-Functor Categories

I have created an nnLab entry: Thomason model category

This is mainly to remind myself of the reference when I have time to come back to it and accordingly just comtains that bit of information at the moment. But maybe you feel like expanding on it a bit more.

Posted by: Urs Schreiber on August 10, 2009 8:10 PM | Permalink | Reply to this

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