A little bit of homological algebra for categorical groups
by
Enrico Vitale
Université catholique de Louvain

Categorical groups, i.e., monoidal groupoids with invertible objects, are a useful tool to study homotopy 2-types, commutative ring theory and low dimensional topological quantum field theory. A structural investigation of the 2-category of categorical groups leads in a natural way to develop a kind of 2-dimensional homological algebra for categorical groups. The aim of this talk is to explain the analogies and the differences between classical homological algebra and its 2-dimensional generalization. For this, we focus our attention on two examples: extensions of categorical groups and the fundamental sequence in non-symmetric cohomology. As far as extensions are concerned, we mainly restrict ourselves to symmetric categorical groups, and we show in what sense a suitably defined symmetric categorical group EXT measures the exactness of Hom. As far as the fundamental cohomology sequence is concerned, we use two different methods. The first one, quite elementary, is based on fibrations of bigroupoids and gives, as further applications, the 2-dimensional analogue of some classical exact sequences in ring theory. The second one is based on the notion of categorical crossed module.

Date received: May 30, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-31.