Commutative semigroup cohomology (Q1176626)

Let \(S\) be a commutative semigroup. A Beck extension of \(S\) by an abelian group object \(A\) of a (comma) category \(\mathfrak L\) consists of a commutative semigroup \(C=(C,q)\) over \(S\), with \(q\) surjective, and for each \(T\in{\mathfrak L}\) a simply transitive abelian group action of \(\hbox{Hom}_{\mathfrak L}(T,A)\) on the set \(\hbox{Hom}_{\mathfrak L}(T,C)\), which is compatible with \(q\) and natural in \(T\). Denote with \({\mathcal H}(S)\) the semigroup \(S\) viewed as a category. A commutative coextension of \(S\) by an abelian group valued functor \(A: x\mapsto A_ x\) on \({\mathcal H}(S)\) consists of a commutative semigroup \(C\), a surjective homomorphism \(p: C\to S\) and left group actions, under which each \(A_ x\) acts simply and transitively on \(C_ x=p^{-1}(x)\), so that \((g\cdot a)b=(\alpha_{x,y}g)\cdot ab\) for all \(x,y\in S\), \(g\in A_ x\), \(a\in C_ x\), \(b\in C_ y\). It is proved that if \(A\) is an abelian group valued functor on \({\mathcal H}(S)\), then the category of Beck extensions of \(S\) by \(A\) is isomorphic to the category of commutative coextensions of \(S\) by \(A\). Triple cohomology for commutative semigroups is described.