Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those additive functors $F:A\to B$ such that the complex $(F(M^i))$ is contractible (in $B$); note that one can certainly construct a universal $F$ satisfying this condition. Did anybody consider functors of this sort (instead of a single $M$ one can certainly consider a collection of complexes here)? One of my questions is: which $A$-complexes become contractible after we apply "the universal" $F$ to their terms?
In one of my papers I have thoroughfully considered the case where $M$ is of length $1$ (i.e., it is a single morphism; this corresponds to a certain localization of $A$); what can one say about the length $2$ case? This case is probably related to exact and derived categories.
