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If $\mathcal{C}$ is a category, then surely the category of simplicial objects $s\mathcal{C}$ is not automatically a model category. What conditions must $\mathcal{C}$ satisfy in order for $s\mathcal{C}$ to have a reasonable model structure?

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2 Answers

up vote 3 down vote accepted

It always has a model structure using Kan's theory of Reedy categories. For a proof, see Hirschhorn Model Categories and their Localizations 15.3.

This is because $\Delta$ and $\Delta^{op}$ are both Reedy categories.

I will address the more general question as well:

If $C$ is any small category, the condition for $M^C$ to be a model category is that $M$ be cofibrantly generated. However, it is not necessarily true that $M^C$ is itself cofibrantly generated. In general, the condition we need for that is called combinatoriality. However, the existence of such a cofibrantly generated model structure that is not combinatorial would disprove Vopenka's principle (a large cardinal axiom), so I cannot give an example of one.

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I stand corrected and informed, thanks! –  Jeff Strom Jan 12 '11 at 18:53
No problem! –  Harry Gindi Jan 12 '11 at 19:00
I thought the questions was: when does the category of simplicial objects in $\mathcal{C}$ have a model structure, without assuming $\mathcal{C}$ is a model category. –  Oscar Randal-Williams Jan 12 '11 at 19:14
I'm tempted to let you think that I asked the smartest possible question, but I didn't. I originally intended $\mathcal{C}$ to be any category. But I think we can see that if $s\mathcal{C}$ has a model structure, then $\mathcal{C}$ must be complete-cocomplete, and then we can use the trivial structure on $\mathcal{C}$ and follow Harry's method. So the question in which we assume a given model category on $\mathcal{C}$ is harder and more interesting, and what was answered. –  Jeff Strom Jan 12 '11 at 19:31
The real question is: what is a "reasonable" model category structure on $s\mathcal{C}$? What constraints should it satisfy? –  Charles Rezk Jan 12 '11 at 21:11
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Quillen gives a couple of sets of sufficient conditions for $s\mathcal{C}$ to be a model category, in his Homotopical Algebra book. Notably, this includes the case when $\mathcal{C}$ is a complete and cocomplete category with a small set of projective generators. This gives model structures for categories of simplicial groups, simplicial rings, simplicial lie algebras, etc.

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Oh, this is interesting! What page is this on? –  Harry Gindi Jan 12 '11 at 19:30
Ah, got it. It's $II.4$. –  Harry Gindi Jan 12 '11 at 19:42
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