Cohomology and \(\mathcal {H}\) -Coextensions

In this chapter we return to the primary motivation for the \(\mathsf {D}\) -cohomology: the structure and classification of \(\mathcal {H}\) -coextensions. Section 4.1 studies the kernel \(\Sigma _p\) of an \(\mathcal {H}\) -coextension \((E,p)\) of a monoid S. The kernel functor \(\Sigma _p:\mathsf {D}(E)\to \mathsf {Gr}\) plays the same role for \((E,p)\) that the kernel subgroup plays for a group coextension. It is our main tool for analyzing \(\mathcal {H}\) -coextensions. Section 4.2 looks at the abelian case where \(\Sigma _p:\mathsf {D}(E)\to \mathsf {Ab}\) , the category of abelian groups. Such coextensions define a full subcategory of the category of \(\mathcal {H}\) -coextensions S, that is analyzed here. In Sect. 4.3 we return to analyze of the general case. We define semifunctors from \(\mathsf {D}(S)\) to the category of groups which along with nonabelian 2-cocycles can be used to construct coextensions S. We give necessary and sufficient conditions for such coextensions to be \(\mathcal {H}\) -coextensions. Cohomological aspects are studied in the fourth section. In particular the third cohomology group comes into play in studying obstructions to coextensions. In a final Sect. 4.5 we use groups as an illustrative example. We show how the classical theorems on the classification of nonabelian extensions of groups by Schreier and Eilenberg-Mac Lane are instances of the results in the previous sections.