Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings

TL;DR

This paper develops a categorical framework linking bimonoidal functors, biextensions, and multilinear functor calculus, providing new insights into the structure of categorical rings and their classification via Mac Lane cohomology.

Contribution

It introduces a nonabelian biextension concept for bimonoidal functors, extends it to multiple variables, and connects these structures to a multilinear functor calculus in bicategories.

Findings

Biextensions satisfy triviality conditions making them bilinear spans.

Multi-extensions can be composed similarly to butterflies.

Classification of ring-like bimonoidal stacks via third Mac Lane cohomology.

Abstract

We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal in each variable, a nonabelian version of a biextension. We show that such a biextension satisfies additional triviality conditions which make it a bilinear analog of the kind of spans known as butterflies and, conversely, these data determine a bimonoidal functor. We extend this result to $n$-variables, and prove that, in a manner analogous to that of butterflies, these multi-extensions can be composed. This is phrased in terms of a multilinear functor calculus in a bicategory. As an application, we study a bimonoidal category or stack, treating the multiplicative structure as a bimonoidal functor with respect to the additive one. In the context of the multilinear functor calculus, we view the bimonoidal structure as an instance of the general notion of pseudo-monoid. We show that when the structure is ring-like, i.e. the pseudo-monoid is a stack whose fibers are categorical rings, we can recover the classification by the third Mac Lane cohomology of a ring with values in a bimodule.