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We associate to a bimonoidal functor, i.e. a bifunctor which is monoidal ineach variable, a nonabelian version of a biextension. We show that such abiextension satisfies additional triviality conditions which make it a bilinearanalog of the kind of spans known as butterflies and, conversely, these datadetermine a bimonoidal functor. We extend this result to $n$-variables, andprove that, in a manner analogous to that of butterflies, thesemulti-extensions can be composed. This is phrased in terms of a multilinearfunctor calculus in a bicategory. As an application, we study a bimonoidalcategory or stack, treating the multiplicative structure as a bimonoidalfunctor with respect to the additive one. In the context of the multilinearfunctor calculus, we view the bimonoidal structure as an instance of thegeneral 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 canrecover the classification by the third Mac Lane cohomology of a ring withvalues in a bimodule.

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