Abstract In the context of abelian groups, Biextensions are biadditive morphisms (Z-bilinear maps) what extensions are homomorphisms. We extend the notion of biextension to monoidal categories (and stacks) with only mild commutativity assumptions. The goal is to classify additive bifunctors of such objects in a way analogous to how one classifies additive functors.
(Equivalently, in this way one obtains a derived bifunctor in the context of the nonabelian derived category.) Applications and motivations stem from the theory of categorical rings and mod 2 phenomena that distinguish various cohomology theories of rings. If time permits we will illustrate some of these applications.