Turkish Journal of Mathematics




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.

First Page


Last Page


Included in

Mathematics Commons