Cohomology of Monoids with Operators

This chapter deals with \(\Gamma \) -monoids, where \(\Gamma \) is a fixed monoid of operators that acts on any given monoid S via a monoid homomorphism \(\Gamma \to \mathrm {End}(S)\) . We provide a cohomology theory for \(\Gamma \) -monoids, say for S, with the groups denoted by \(H^n_\Gamma (S,\mathcal {A})\) . When both \(\Gamma \) and S are groups, this theory goes back to that first introduced by J.H.C. Whitehead in his seminal 1950 paper on the cohomology of groups with operators. If \(\Gamma =1\) is trivial, then the cohomology groups \(H^n_\Gamma (S,\mathcal {A})\) are just the ordinary \(\mathsf {D}\) -cohomology groups \(H^n(S,\mathcal {A})\) . Computation by cocycles, connections with other known cohomology theories and applications to the classification of equivariant coextensions of monoids with operators are topics discussed in this chapter.