I'm reading Voevodsky and Morel's book '$\mathbb{A}^1$-homotopy theory of schemes'. In Remark 3.1.15, it says that for any simplicial fibrant sheaf $F$ and open sets $U\subseteq V$, $F(V)\to F(U)$ is a fibration.


Prove by definition. We have a bifunctor
$$\begin{array}{ccccc}sSet&\times&Shv(Sm/k)&\to&sShv(Sm/k)\\(S&,&F)&\mapsto&S\times F\end{array},$$
where $(S\times F)(X)\_n=S\_n\times F(X)$. Consider the coequalizer
$$\Lambda^n\_k\times U\rightrightarrows\Lambda^n\_k\times V\coprod\triangle^n\times U\to C.$$
Then there is a map $i:C\to \triangle^n\times V$ and the question is reduced to the RLP of $F$ w.r.t $i$. So I want to prove $i$ is a trivial cofibration.


It's obviously a cofibration but I'm stuck at proving it's a weak equivalence. It suffices to prove that the functor $-\times F:sSet\to sShv(Sm/k)$ is a left Quillen functor since we could then use the pushout diagram of $C$. So we are going to prove that trivial cofibrations commute with infinite products, by passing to stalks...


I think we have to prove that geometric realization functor commutes with infinite products, as least up to a weak equvalence. Is this true?


Thanks a lot!



Do infinite products commute with trivial cofibrations, for simplicial sets?
This fact admits a much easier proof.
To show that for any simplicial fibrant sheaf F and open sets U⊆V the map F(V)→F(U) is a fibration
it suffices to show that F(V)→F(U) has a right lifting property with respect to
horn inclusions.


Since F(V)→F(U) can be rewritten as Map(U→V,F),
we can move U→V using the two-variable adjunction between ⨯ and Map,
which boils down the problem to showing that F→1
has a right lifting property with respect to the pushout product
of U→V and a horn inclusion.


Since F→1 is a fibration of simplicial sheaves
and the pushout product axiom is satisfied for the categorical product,
it suffices to observe that U→V is a cofibration (i.e., a monomorphism)
of simplicial sheaves and a horn inclusion is an acyclic cofibration of simplicial sets.


(By the way, the geometric realization functor does not commute with infinite products,
nor is this true if we derive both of them.)


