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Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see below for definition). Is there a closed model category structure on $Com^+(A)$ such that weak equivalences are quasi-isomorphisms and $R$ is (or contains) the class of cofibrant objects?

Here is a definition of class of objects adapted to a left exact functor $F: A \to B$ (from Gelfand and Manin): it is a class of objects $R$ stable under direct sums, $F$ maps acyclic complexes from $Com^+(R)$ to acyclic ones and each object is in $A$ is a subobject of an object in $R$


(1) This question originated from this one:

(2) This question does not really have anything to do with algebraic geometry, but I thought that algebraic geometers may have something to say about this.

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I added the model categories tag. There are many people here who know about such things, and I hope you get a good answer. Sadly, my knowledge was used up in the comments I made to your earlier post. – David White Jun 29 '11 at 3:11
Thanks David, I hope so – Mikhail Gudim Jun 29 '11 at 4:18
for sheaves (as cats) and tensor product (or pullbacks) (as your functor) there is a flat model structure. (paper by hovey?) In general I have no idea. – babubba Jun 29 '11 at 8:02

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