Abstract

In the context of abelian groups, multiextensions are to multilinear maps what extensions are to homomorphisms. Multiextensions can be defined for categories and stacks, where they classify functors which are additive in each variable. Applications and motivations stem from the problem of describing various kinds of categorical rings and the cohomology theories that arise as their characteristic classes, such as those of Hochschild and Mac Lane.

In this part we'll look at how categorical rings realize MacLane cohomology classes. Then we'll introduce multiextensions and show how they help efficiently package the notion of categorical ring. Finally we show how this allows to quickly recover the ring cohomology theories previously examined.