# On weight complexes, pure functors, and detecting weights

## TL;DR

This paper studies weight complexes and pure functors in triangulated categories, introducing new tools for detecting weights and applying these concepts to equivariant stable homotopy and motivic categories.

## Contribution

It introduces pure (co)homological functors that ignore all non-zero weights and provides new weight structures, enhancing the detection of weights in triangulated categories.

## Key findings

- Pure functors are described via weight complexes.
- Pure cohomological functors correspond to Bredon cohomology in $SH(G)$.
- Certain functors are proven to be conservative and detect 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 also give some (more) new weight structures. Moreover, we prove that certain exact functors are conservative and "detect weights".