Higher cohomologies for presheaves of commutative monoids

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 H-coextensions of presheaves of commutative regular monoids

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