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this is my first thread on mathoverflow, and apologies if this is a trivial question.

In Dwyer and Spalinski's Homotopy theories and model categories, they gave the definition of derived functors, and based on their definition, we require the source C of a functor F : C $\to$ D to be a model category, and if a derived functor (no matter left or right) exists, denoted as DF, then

DF : Ho(C) $\to$ D

The notes also gives a sufficient condition of the existence of left derived functor, but because of duality, we can rephrase the same for right derived functor: if the functor F is as above, and F(f) is an isomorphism whenever f is a weak equivalence between fibrant objects in C, then the right derived functor (RF, t) of F exists.

I think I accept the concepts above fairly well, but then in Section 3.1 of Hartshorne's Algebraic Geometry, he introduced the concept of derived functors of covariant left exact functors F : A $\to$ B, where A and B are abelian categories, with A having enough injectives. Though Hartshorne didn't mention the condition of existence, he wrote if F: A $\to$ B is a covariant left exact functor between abelian categories, and A has enough injectives, then we construct the right derived functor...

So, does the concepts in the two scenarios above correspond? For example (this might be very wrong, just my guess), does "an abelian category with enough injectives" automatically have a model structure?

If the answer is "yes" and we have a model structure compatible with the cohomology objects, does "a covariant left exact functor" correspond to "a functor sending fibrant objects to isomorphisms", and do other properties of derived functors, such as the universal property, also hold?

If the answer is "no", then does it mean we can construct derived functors without model structure and the associated homotopy category? If so, how? Also, are there any relationships between model categories and abelian categories?

Thank you very much for reading my long and tedious thread. I appreciate any insights.

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    $\begingroup$ You have two questions — the general one in the title, and the more specific one in the body about whether the abelian category case can be viewed model-categorically. I guess the latter is your real question, so just a quick note here on the former: there are various well-strudied settings a little weaker than model categories which nevertheless suffice for constructing derived functors, e.g. semi-model categories. $\endgroup$ – Peter LeFanu Lumsdaine Feb 6 '17 at 8:47
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    $\begingroup$ An abelian category $\mathcal{A}$ defines a derived category $D(\mathcal{A})$, that is the localisation of the category of cochain complexes in $\mathcal{A}$ with respect to quasi-isomorphisms. The derived functors of exact functors between abelian categories are induced functors between the derived categories, although this is not how they were historically introduced. The category that admits the desired model structure, making the two notion coinicde, is the category of (co)chain complexes, see this nLab article. $\endgroup$ – user337830 Feb 6 '17 at 10:24
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    $\begingroup$ It might also help to know that injective and projective resolutions are particular fibrant and cofibrant replacements, on the category of (co)chain complexes of an abelian category with enough injective or projective objects, respectively. $\endgroup$ – user337830 Feb 6 '17 at 10:31
  • $\begingroup$ I'm not sure if this really addresses the question but it sounds like Local and Stable Homological Algebra in Grothendieck Model Categories by Cisinski and Déglise might be relevant. $\endgroup$ – Karol Szumiło Feb 6 '17 at 21:27
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This is not quite the answer to your question as you pose it. I hope it will be useful anyway. By and large I am just expanding user337830 comments. Everything will use homological grading (what can I say, I am a homotopy theorist :)).

The relationship between derived functors in homological algebra and derived functors in homotopy theory is very close. In fact the notion of derived functor in homotopy theory is a (very successful) attempt to generalize the homological algebra notion.

Let me talk about right derived functors for one second. Let $A$ be a Grothendieck abelian category. Then it is known it has enough injectives. This allows us to define a model structure on the category $Ch(A)$ of chain complexes in $A$ such that

  • weak equivalences are quasi-isomorphisms;
  • cofibrations are chain maps $C_*\to D_*$ that are levelwise injective;
  • fibrations are chain maps $C_*\to D_*$ that are levelwise surjective and such that the kernel $K_*$ is a dg-injective complex (that is all $K_n$ are injective objects of $A$ and for all acyclic complexes $E_*$ every chain map $E_*\to K_*$ is nullhomotopic, bounded above complexes of injectives are an example of such).

This model structure is sometimes called the injective model structure on $Ch(A)$.

In particular if $M\in A$, an injective resolution of $M$ gives a fibrant replacement for the chain complex consisting of $M$ in degree 0. So it is easy to see that if $A,A'$ are two Grothendieck abelian categories and $F:A\to A'$ is a left-exact functor, the derived functor of $F$ is precisely the right-derived functor of $F_*(-):Ch(A)\to Ch(A')$.

Similarly if $A$ has enough projectives you can form a projective model structure that will allow you to describe left derived functors of right-exact functors.

Now, these two model structures in fact present the same homotopy theory (e.g. they have the same weak equivalences! More to the point, they are Quillen equivalent), so you can pass from one to the other with a minimum of fuss. Their common homotopy category is called the (unbounded) derived category of $A$.

There are also other model structures on $Ch(A)$ with the same weak equivalences that are more convenient for other purposes, e.g. the flat model structure (where the fibrant objects are complexes of flat modules $C_*$ such that $C_*\otimes-$ preserves quasi-isomorphisms). In general there is a correspondence between algebraic model structures on $Ch(A)$ and cotorsion pairs on $A$. See this paper by Hovey for more details.

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