Relating Brauer categories, Koszul complexes, and graph complexes

The purpose of this paper is to investigate the relationship between hairy graph complexes associated to cyclic operads and their counterparts for operads (and, more generally, dioperads). This is based on the author's interpretation of these as Koszul complexes for the associated modules over the respective appropriate twisted downward (walled) Brauer category. The general question of relating such Koszul complexes is addressed by analysing the relationships between the respective twisted Brauer-type categories, proceeding through a direct analysis. The passage from the walled to unwalled context involves functors induced by the disjoint union of finite sets. As an application, for the cyclic operad associated to an operad, this leads to an explicit relation between the respective (hairy) graph homologies.