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?



Model categories of simplicial objects
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.


