# Derived functors commute with filtered colimits?

I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer.

So I'd actually like to know what the definite answer is. The context is the following : we have $$\mathbf{AB5}$$ abelian categories $$A,B$$, and a right exact functor $$F:A\to B$$. Assume $$A$$ has enough projectives, so that we can make sense of the left derived functors of $$F$$. We wish to know when those left-derived functors preserve the filtered (or directed) colimits $$I\to A$$ that $$F$$ preserves. An obvious necessary condition is that colimits of projectives in $$A$$ are $$F$$-acyclic.

Now if $$A$$ is concrete enough and really nice, we can make "projective resolution" into a functor $$A\to Ch^+(A)$$; but in the general case, the "fundamental lemma" that is mentioned in the answer only gives a well-defined functor $$A\to K^+(A)$$. If there is a "concrete" projective resolution, then I understand the answer with no problem, but I have trouble seeing how to get from there to the desired result, because there does not seem to always be a functorial projective resolution with values in $$Ch^+(A)$$.

It seems to me if we have a directed system $$(C_i)_{i\in I}$$, take projective resolutions $$(P_{\bullet, i})_{i\in I}$$, compute for each $$i\leq j$$ the (unique up to homotopy) induced chain map $$P_{\bullet, i}\to P_{\bullet, j}$$, and try to lift this system of chain maps to a functor $$I\to Ch^+(A)$$, we run into some "cohomological obstruction" (cohomology of $$I$$ perhaps, it would be interesting if it were actually related to some (co)homology of $$I$$)

Indeed, what seems to happen is that, denoting $$f_{ji}$$ a (fixed, concrete) chain map $$P_{\bullet, i }\to P_{\bullet, j}$$ extending $$C_i\to C_j$$ whenever $$i\leq j$$, we get equations like $$f_{kj}f_{ji} - f_{ki} = dh_{ijk}+h_{ijk}d$$ for $$i\leq j \leq k$$ and homotopies $$h_{ijk}$$ and we're looking for $$g_{ij}$$'s such that $$g_{ij}-f_{ij} = d h'_{ij} + h'_{ij}d$$ and $$g_{kj}g_{ji} - g_{ki} = 0$$.

I can't seem to make this intuition precise, but this reminds me a lot of group cohomology, and it looks like there could be some cohomology of $$I$$ lurking behind all this (but I can't make this precise, because the $$f_{ij}$$'s don't seem to be a cocyle looking to be a coboundary, as their "differential" -probably something like $$f_{kj}f_{ji}-f_{ki}$$ - is not $$0$$, but a coboundary)

So this digression makes me ask a few (related) questions in the end :

(1) How do we resolve the problem I raised in the mentioned answer/how do you prove that under the given conditions, left derived functors preserve filtered (directed if it's easier) colimits ?

(2) Is the given answer actually valid, or does it depend on some more conditions on $$I$$ ?

(3) If the answer to (2) is that it depends on $$I$$, can the conditions on $$I$$ be made "cohomological", in a sense that makes precise the vague intuition I gave at the end of my question ? How ?

(4) If the answer to (2) is that it depends on $$I$$, and the answer to (3) is that it does depend on some sort of cohomology of $$I$$, are there more studies on how "cohomology" of indexing categories $$I$$ influences functor categories $$C^I$$, for $$C$$ abelian, or more generally ?