Resolutions, relation modules and Schur multipliers for categories

Abstract

We show that the construction in group cohomology of the Gruenberg resolution associated to a free presentation and the resulting relation module can be copied in the context of representations of categories. We establish five-term exact sequences in the cohomology of categories and go on to show that the Schur multiplier of the category has properties which generalize those of the Schur multiplier of a group.

Cohomology and extensions of ordered groupoids

“…Related work on the cohomology of categories can be found in [29,30,32]. In particular, Webb [30] obtains five-term exact sequences in the (co)homology of categories.…”

“…Related work on the cohomology of categories can be found in [29,30,32]. In particular, Webb [30] obtains five-term exact sequences in the (co)homology of categories. However, since the constructions in this paper originate from an ordered groupoid G rather than from its associated category Ł(G), there are resulting differences at key points, such as in the definition of the analogue of the group ring, of the augmentation ideal, and the notion of extension.…”