Let $ F:\mathcal A\to \mathcal B$ be an additive right exact functor between abelian categories $ \mathcal A$ and $ \mathcal B$ . Then for a short exact sequence $ $ 0\to A\to B\to C\to 0 $ $ in $ \mathcal A$ there is a long exact sequence of derived functors of $ F$ : $ $ \dots \to L_1 F(B)\to L_1 F(C)\to F(A)\to F(B)\to F(C)\to 0 $ $
I wonder what can we say in a more general case when $ \mathcal A$ is a model category and $ \mathcal B$ is a stable model category and $ F$ be a left Quillen functor. In practice I mostly care about the case when these categories are categories of simplicial objects and $ \mathcal B$ admits Dold-Kan correspondence (i.e. equivalent to bounded below chain complexes in some abelian category). If we start with cofiber sequence $ $ A\to B\to C $ $ of course, we have a long exact sequence $ $ \dots \pi_1 LF(C)\to F(A) \to F(B) \to F(C) \to 0 $ $
but how much can we tell for a fiber sequence $ A\to B \to C$ ? Intuitively, there may be some (Leray-Serre like ?) spectral sequence instead of a long exact sequence. Thx.
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