Model categories of simplicial objects 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?
 A: 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.
A: 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.
