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In this short paper by G. Maltsiniotis derived functors are presented as Kan extensions along the localization functor.

I began studying derived categories only a couple of months ago, so I'm not at all an expert, but this approach is completely new to me, as I've not found it in any of the book in my hand (the most advanced is Categories and Sheaves but I've never read it sistematically, so maybe I haven't noticed it there).

Can you give, if it exists, a reference which rewrites some of the basics on derived functors taking into account the "Kan extensions perspective"?

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I realize I'm late to the party but I think you'd enjoy Emily Riehl's notes "CATEGORICAL HOMOTOPY THEORY". –  Dan Petersen Apr 2 '12 at 11:58
    
Not at all, thank you! –  tetrapharmakon Apr 3 '12 at 23:49
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In the context of model categories, the left (reps. right) derived functors are a generalization of the traditional notion of a derived functor. The left (resp. right) derived functor $LF$ of a functor $F\colon \mathbf C\to \mathbf D$ where $\mathbf C$ is a model category is exactly a right (resp. left) Kan extension along the localization $\gamma\colon \mathbf C\to\operatorname{Ho}(\mathbf C)$. This terminology of derived functors stems from what happens in the category of chain complexes. In general, computing $LF(X)$ is computing $F(QX)$ where $QX$ is a cofibrant replacement of $X$ where we view an $R$-module $X$ as a chain complex concentrated in degree 0 (the definition of $LF$ is unique up to natural isomorphism). In the category of chain complexes, $QX$ is a projective resolution of $X$, and hence the homology of $LF(X)$ for a right exact functor $F$ is the left derived functors of $F$. For example, if $F$ is $-\otimes_R A$, then the homology of $LF$ will be $\operatorname{Tor}_*^R(-,A)$.

We have a similar story for right derived functors (which are left Kan extensions along the localization functor). Here, $RF(X)=F(RX)$ where $RX$ is a fibrant replacement. For chain complexes, this is an injective resolution, and so the cohomology of $RF(X)$ is the right derived functors of $F$.

If you are unfamiliar with model categories, I'd recommend Homotopy theories and model categories by Dwyer and Spalinski. In particular, example 9.6 deals with what I just outlined.

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Clear and useful, thanks! A single detail: when you write $\gamma\colon \mathbf C\to Ho(\mathbf C)$ you are localizing wrt weak equivalences, aren't you? –  tetrapharmakon Dec 27 '11 at 18:34
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Correct. The nice part about working with model categories is that we get an explicit construction of $\operatorname{Ho}(\mathbf C)$, so describing the localization functor is very easy to do, and we don't run into the set theoretic issues we otherwise would. –  SL2 Dec 27 '11 at 18:40
    
@SL2. Maybe your $L(QX)$ in the fifth line should be $F(QX)$? –  a.r. Dec 27 '11 at 18:54
    
@SL2: maybe you are wrong in saying "The left (resp. right) derived functor LF [...] is exactly a left (resp. right) Kan extension": Maltsiniotis' paper says that the right derived functor of $F$ is the left Kan extension along localization. –  tetrapharmakon Dec 27 '11 at 18:56
    
@AgustíRoig Fixed, thanks. –  SL2 Dec 27 '11 at 19:07
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I cannot comment so I post this as an answer: derived functors are kan extensions via definition 13.3.1, 7.3.1 and 2.3.2 of Kashiwara and Shapira "Categories and Sheaves" ( well, they do not use the words kan extension in 2.3.2).

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