Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings
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 -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.
Cite
@article{arxiv.1501.04664,
title = {Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings},
author = {Ettore Aldrovandi},
journal= {arXiv preprint arXiv:1501.04664},
year = {2017}
}
Comments
Accepted version to appear in Theory and Applications of Categories; 61 Pages; the new Appendix E contains the full hypercohomology computation of the characteristic class of a ring-like stack