On weight complexes, pure functors, and detecting weights

Abstract

This paper is dedicated to the study of weight complexes (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that "ignore all non-zero weights"; these have a nice description in terms of weight complexes. For the weight structure w G generated by the orbit category in the G-equivariant stable homotopy category SH(G) the corresponding pure cohomological functors into abelian groups are the Bredon cohomology associated to Mackey functors ones; pure functors related to motivic weight structures are also quite useful. Our results give some new weight structures and the conservativity of certain weight-exact functors; so they can be used to prove that a theorem of J. Ayoub implies the conservativity of realizations of Q-linear geometric motives over characteristic 0 fields. We also prove that certain functors "detect weights", i.e., check whether an object belongs to the given level of the weight filtration.