Cohomology of Presheaves of Monoids
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CohomologyPresheaf of monoidsMonoidal prestackSimplicial setHomotopy colimit
Carrasco, P.; Cegarra, A.M. Cohomology of Presheaves of Monoids. Mathematics 2020, 8, 116. [doi:10.3390/math8010116]
The purpose of this work is to extend Leech cohomology for monoids (and so Eilenberg-Mac Lane cohomology of groups) to presheaves of monoids on an arbitrary small category. The main result states and proves a cohomological classification of monoidal prestacks on a category with values in groupoids with abelian isotropy groups. The paper also includes a cohomological classification for extensions of presheaves of monoids, which is useful to the study of H-extensions of presheaves of regular monoids. The results apply directly in several settings such as presheaves of monoids on a topological space, simplicial monoids, presheaves of simplicial monoids on a topological space, monoids or simplicial monoids on which a fixed monoid or group acts, and so forth.