# is this a simplicial model category?

A basic result in the theory of model categories is that simplicial sets form a simplicial model category. The same is true for simplicial $k$-algebras. I have two questions related to this. One involves defining model categories via gluing and the second involves defining model categories via functor of points. I have no idea if these questions are extremely easy or difficult. I am sure they display my lack of knowledge when it comes to model categories.

Consider the category $\mathbb{V}_k$ of all separated schemes of finite type over a field $k$ along with a grothendieck topology $\mathcal{T}$ on $\mathbb{V}_k$.

Question A) Is the simplicial cateogry $s\mathbb{V}_k$ of $\mathbb{V}_k$ a simplicial model category (or even just a model category)?

Question B) Let $\mathbb{S}_k$ be the category formed by subfunctors of representable functors from $\mathbb{V}_k \to Sets$. Let $s\mathbb{S}_k$ be the simplicial category of $\mathbb{S}_k$. Is $s\mathbb{S}_k$ a simplicial model category?

-
Any category with finite (or small, to taste) limits and colimits is automatically a model category in a trivial way. You should at least specify what kind of weak equivalences you wish to consider. –  Zhen Lin Sep 19 '13 at 22:17
"Is $X$ a model category" is a type error. Being a model category is a structure, not a property. –  Qiaochu Yuan Sep 19 '13 at 22:44
The weak equivalences should be the ones induced from the weak equivalences in the category of simplicial $k$-algebras. –  Andrew Stout Sep 19 '13 at 23:32
Regarding A you might have a look at Quillen's Homotopical Algebra, Section II.4. There are stated general conditions when simplicial objects form a simplicial model category. –  Lennart Meier Sep 19 '13 at 23:43
Prof. Meier: Thanks! I will try to prove that it satisfies these conditions. –  Andrew Stout Sep 20 '13 at 0:24