Higher cohomologies for presheaves of commutative monoids

Authors: PILAR CARRASCO, ANTONIO M. CEGARRA

Abstract: We present an extension of the classical Eilenberg-MacLane higher order cohomology theories of abelian groups to presheaves of commutative monoids (and of abelian groups, then) over an arbitrary small category. These high-level cohomologies enjoy many desirable properties and the paper aims to explore them. The results apply directly in several settings such as presheaves of commutative monoids on a topological space, simplicial commutative monoids, presheaves of simplicial commutative monoids on a topological space, commutative monoids or simplicial commutative monoids on which a fixed monoid or group acts, and so forth. As a main application, we state and prove a precise cohomological classification both for braided and symmetric monoidal fibred categories whose fibres are abelian groupoids. The paper also includes a classification for extensions of commutative group coextensions of presheaves of commutative monoids, which is relevant to the study of $\mathcal{H}$-coextensions of presheaves of commutative regular monoids.

Keywords: Commutative monoid, simplicial set, presheaf, cohomology, extension, Schutzenberger kernel, fibration, monoidal category, braiding, symmetry

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