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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.

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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.

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    $\begingroup$ I definitely agree with the first line. but I don't quite see the relation between being fibrant in the local projective model structure and having trivial sheaf cohomology. $\endgroup$
    – Simon Henry
    Commented Feb 27, 2020 at 19:29
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    $\begingroup$ @SimonHenry: I added a specific example. $\endgroup$
    – Dmitri Pavlov
    Commented Feb 27, 2020 at 19:58
  • $\begingroup$ I mean the injective model structure. That is why I want to find a system of generators. $\endgroup$
    – Nanjun Yang
    Commented Feb 27, 2020 at 23:19
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    $\begingroup$ @NanjunYang: The generators you wrote down are precisely the generating cofibrations for the projective and local projective model structures. The prevailing opinion with respect to the generating cofibrations for the injective model structure is that there is no explicit such a set of generators, excluding some special categories of presheaves (but Sm/k is not one of them). $\endgroup$
    – Dmitri Pavlov
    Commented Feb 28, 2020 at 4:07
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    $\begingroup$ @NanjunYang: In the motivic context, you would need K(A,n)=Γ(A[n]) to be locally injectively fibrant and not just injectively fibrant. The argument in my post shows that K(A,n)=Γ(A[n]) is not a local object, so cannot be locally injectively fibrant. Even ignoring locality, though, I am pretty sure that K(A,n)=Γ(A[n]) is not injectively fibrant. Why not use the projective local model structure instead, where the Eilenberg–MacLane spectra seem to be fibrant? $\endgroup$
    – Dmitri Pavlov
    Commented Feb 28, 2020 at 15:46

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