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A basic result of homological algebra says that if $\mathsf A$ is an abelian category with enough projectives, then the mapping $P:\mathsf{Obj}(\mathsf A)\rightarrow \mathsf{Obj}(\mathsf{K} ^+(\mathsf A))$ sending each object to the homotopy type of its projective resolutions almost lifts to a functor in the sense that $P(g\circ f)\cong P(g)\circ P(f)$.

  1. Is there a general way to fix the "almost" functoriality of $P$? If not, are classical derived not actually functors?

In Riehl's Categorical Homotopy Theory, sec 2.3, the author says that in $R$-$\mathsf Mod$, a more refined process enables the replacement of any complex with a quasi-isomorphic complex of projectives. She then says that careful choice of projective resolution yields a proper functor $\mathsf{Ch}_\bullet ^+(R)\rightarrow \mathsf{Ch}_\bullet ^+(R)$ along with a natural quasi-isomorphism $q:Q\Rightarrow 1$. In other words, projective replacement is a left deformation of the homotopical category $\mathsf{Ch}_\bullet ^+(R)$ with q.i's as weak equivalences.

  1. What is this "refined process" and "careful choice"?
  2. What is a general condition on an abelian category with enough projectives that would ensure projective replacement become functorial and moreover a left deformation as in $R$-$\mathsf{Mod}$?

In Remark 2.3.2, the author notes the absense of functorial deformations, the usual process of projective replacement defines total derived functors, but not point-set level derived functors.

  1. What is meant by functorial deformations?

Update: This question seems to be related, but I don't see any conditions guaranteeing a "projective resolvent functor".

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    $\begingroup$ 1. Classical derived functors are functors – between the derived categories. 2. Look up Cartan–Eilenberg resolutions. 3. A functorial projective replacement should be enough. 4. There is only one possible meaning. $\endgroup$ – Zhen Lin Aug 11 '15 at 22:40
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    $\begingroup$ @ZhenLin regarding (2), one of the first googles is the stacks project. There it proven in particular that projective replacement exists. In remark 13.21.4 it is said that this replacement is functorial, but I don't see why this is true at all. (I don't know anything about spectral sequences.) Regarding (3), I was trying to ask for conditions on an abelian category that would ensure functoriality. On another note, if you would post your comment as a more detailed answer I'd gladly accept it. $\endgroup$ – Arrow Aug 12 '15 at 12:58
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  1. Derived functors are functors. What is your confusion? You divide out the chain homotopy relation in the target category.

2,3. In order to construct a functorial projective resolution you need to have for each object A in the abelian category a natural surjection from a projective object $P_A \to A$.

  1. This is similar to functorial cofibrant replacements.

In most practical situations the functoriality exists. I'll give you one example where it does not: consider the abelian category of finitely generated abelian groups, then there enough projectives, but there is no way to choose them functorially.

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