Let $X$ be a topological space and $F$ a flat sheaf of abelian groups. It is well known that taking the Godement resolution of $F$ gives rise to a "flabbyflat" or "flasqueflat" resolution, in the sense, that it is a complex where ever sheaf is flabby and flat at the same time.
Is there an analogue for the unbounded case? Specifically I wonder if using the definitions in Spaltensteins paper (Resolutions of unbounded complexes), one can take a K-flat complex and then take some (functorial?) resolution to obtain a complex which is K-flabby but also still K-flat.
I think that some finiteness conditions are needed and I am totally fine with this.
