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

##
Ettore Aldrovandi

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.

Keywords:
Categorical ring, biextension, bimonoidal, ring-like stack, butterfly,
multi-extension, multi-category, multi-functor, Mac Lane cohomology

2010 MSC:
18D10, 13D03, 18G55, 55P43, 14A20

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 27, pp 889-969.

Published 2017-08-02.

http://www.tac.mta.ca/tac/volumes/32/27/32-27.pdf

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