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Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings. (English) Zbl 1429.18006

Summary: 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.

MSC:

18M50 Bimonoidal, skew-monoidal, duoidal categories
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
18N40 Homotopical algebra, Quillen model categories, derivators
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
14A20 Generalizations (algebraic spaces, stacks)
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References:

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