Resolutions by Adapted Class of Objects and Model Categories

My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) I've just started learning these things, so my question may be trivial.

When one looks at the model category of (non-negatively graded) chain complexes, the procedure of taking cofibrant replacement is taking projective resolution of a given complex (Cartan-Eilenberg resolution). This gives a standard recipe for computing derived functor via projective resolution.

However, it is a fact that in order to compute derived functors of an additive left exact functor $F: A \to B$ one can take resolution by any adapted to $F$ class of objects $R$. Actually, there is an equivalence (induced by inclusion) between the derived categories of (non-negatively graded) complexes of objects from $R$ and the derived category of $A$. Class of injective (projective) objects is adapted to any functor. (See Gelfand and Manin III.6 for example.)

How can one prove these facts in the language of model categories?

Mikhail

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I had to look this up, so I figured I'd comment to help others. Adapted Object: ncatlab.org/nlab/show/class+of+adapted+objects – David White Jun 27 '11 at 23:14
Just to confirm, you're looking at $Ch(\mathcal{A})$ where $\mathcal{A}$ is an abelian category with enough projectives and injectives? Based on the article I linked above, those seem to be prerequisites for your fact about adapted objects being used to compute derived functors. I don't think a model category $\mathcal{C}$ needs to have enough projectives. In general cofibrant replacement is not the same as projective resolution. See e.g. mathoverflow.net/questions/10246/… – David White Jun 27 '11 at 23:35
This doesn't answer your question, but it's a comment that often it's better to define derived functors so they only depend on $\mathcal{C}$ not on a choice of cofibrant replacement. I suppose your way would allow easier computation but might lose a lot by making that choice. A good reference is this page of Hovey's book: books.google.com/… – David White Jun 27 '11 at 23:44
Here is the definition of class of objects $R$ adapted to a left exact functor $F:A \to B$ (from Gelfand and Manin): $R$ is closed under finite direct sums, $F$ maps acyclic complex from $Com^+(R)$ into an acyclic complex, any object from $A$ is a subobject of an object from $R$. – Mikhail Gudim Jun 28 '11 at 1:55
Maybe one should ask this question instead: 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 $F$. 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. If the answer to this question is "yes", then I guess this gives an answer to the original question. But maybe the original question has other solutions...? – Mikhail Gudim Jun 28 '11 at 2:15

(Disclaimer: I don't have a copy of Gelfand and Manin to hand, so I'm only conjecturing that what I'm about to say is relevant to your question.)

I think what you might be looking for is the construction of derived functors via deformations. This is a generalization of the construction that's used in the context of model categories to so-called homotopical categories, due originally to Dwyer, Kan, Hirschhorn, and Smith. A good summary can be found in the first few sections of this paper.

The main idea is the following: if $F \colon C \to D$ is a functor between categories equipped with some notion of weak equivalences satisfying the 2-of-3 property, to construct a left derived functor of $F$, you don't need a full model structure on $C$. Instead it suffices to have an endofunctor $Q$ equipped with a natural weak equivalence $Q \Rightarrow 1_C$ such that $F$ preserves all weak equivalences between objects in the image of $Q$.

The theorem is that in this case $LF = FQ$ together with the natural transformation $FQ \Rightarrow F$ is a (point-set) left derived functor of $F$ (meaning, if you compose with the localization functor $D \to Ho(D)$, this becomes a left derived functor in the usual sense). It's really easy to prove once you know it's true. I suggest it as an exercise.

The notation is meant to suggest cofibrant replacement. Assuming we have a functorial cofibration - trivial fibration factorization, we can factor maps $\emptyset \to X$ to obtain a cofibrant object $QX$ and a natural weak equivalence $QX \to X$. If $F$ is left Quillen (preserves (trivial) cofibrations and the initial object), then $F$ preserves all trivial cofibrations between cofibrant objects. By Ken Brown's lemma, $F$ then preserves all weak equivalences between cofibrant objects. So the left derived functor of $F$ can be constructed simply by precomposing with some cofibrant replacement.

But the point is it doesn't matter what sort of cofibrant replacement we use, or even that it is a cofibrant replacement for some model structure. Maybe this is what's going on with the class of objects adapted to $F$.''

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This looks exactly right to me. In the case at hand Q should be the "replace by adapted class resolution" functor. – Chris Schommer-Pries Dec 23 '11 at 19:47
Should we always expect that we can take strictly functorial replacements with respect to a class of objects adapted to $F$? – Justin Noel Dec 24 '11 at 7:52
@ Justin. You are right. I don't think we should expect there to exist such a Q as a functor at the level of C, just as we should not expect there to exist "point-set level" derived functors. However I think there is a remedy for this. (I'm just speculating right now, if you know a better answer I'd love to hear it). I think there does always exist a functor Q from C into the full sub-simplicial category of the Hammock localization of C spanned by the adapted resolutions. Moreover F is homotopical when restricted to this subcategory, so extends to a map from the ... (cont) – Chris Schommer-Pries Dec 24 '11 at 20:36
(cont). ... F extends to a simplicial functor from the Hammock localization of the sub-category of "adapted resolutions" of C into the hammock localization of D. The composite FQ isn't a point-set level derived functor, but it should be a good analog when the point set level derived functor might not exist or when you can't get an honestly functorial Q. – Chris Schommer-Pries Dec 24 '11 at 20:39