I want to show that simplicial abelian sheaves are fibrant. For this, I wonder whether a morphism between simplicial sheaves is a fibration iff it has RLP w.r.t. all morphisms like
$$\Lambda^n\_k\times X\to\triangle^n\times X$$
where $X\in Sm/k$. Is this claim true?


Here weak equivalences are stalkwise weak equivalences and cofibrations are monomorphisms.



Are simplicial abelian sheaves fibrant?
Fibrant in what model structure?


Simplicial abelian sheaves (and presheaves) are fibrant
in the projective model structure because
all simplicial abelian groups are fibrant.


Simplicial abelian sheaves are definitely
not fibrant in the local projective model structure
because sheaf cohomology groups can be nontrivial.
For a specific example, consider the simplicial abelian sheaf
Γ(Z[1]) on the site of smooth manifolds,
given by applying the Dold-Kan functor Γ
to the chain complex of sheaves Z[1] given by placing
the constant sheaf Z in chain degree 1.
If the simplicial abelian sheaf Γ(Z[1]) was fibrant in the local projective model structure,
than the sheaf cohomology of a smooth manifold M
in degree 1 with coefficients in Z
could be computed by evaluating Γ(Z[1]) on M
and then taking the homology group in degree 0.
However, the homology group of Γ(Z[1])(M) in degree 0 is 0 for any M,
a contradiction with the case M=S^1, for which H^1(S^1,Z)=Z.


