Other Cohomologies

The chapter begins by comparing the \(\mathsf {D}\) -cohomology of a monoid S with two earlier cohomology theories of S: the Eilenberg-Mac Lane cohomology and the Hochschild-Mitchell cohomology. Bringing the \(\mathsf {D}(S)\) category into the picture we find that both earlier cohomologies can be seen as special cases of the DD-cohomology for any given monoid. As far as cohomological dimensions go, for any given monoid S, one has: \(dim_{_{\mathrm {EM}}}(S)\leq dim_{_{\mathrm {HM}}}S\leq \mathsf {D} im (S)\) . Special cases are discussed. Next, a canonical homomorphic image of \(\mathsf {D}(S)\) that is important in the study of \(\mathcal {H}\) -coextensions is introduced. In general \(\mathsf {D}(S)\) and \(\mathcal {D}(S)\) do not induce equivalent cohomology theories. But \(\mathcal {D}(S)\) -cohomology does become in essence a sub-cohomology of \(\mathsf {D}(S)\) -cohomology via covariant means for the elegant class of inverse monoids. Indeed, for inverse monoids, the \(\mathsf {D}\) -cohomology for coefficients factoring through \(\mathcal {D}(S)\) agrees with the \(\mathcal {D}\) -cohomology of their corresponding \(\mathcal {D}\) -functors, leading to a agreement with the cohomologies of M. Loganathan and H. Lausch given for inverse semigroups in the monoid case. Next, there are three sections devoted to cohomologies that properly include the \(\mathsf {D}\) -cohomology of monoids. Section 3.5 shows how \(\mathsf {D}\) -cohomology is a particular instance of the cohomology of simplicial sets given by Gabriel and Zisman. Section 3.7 is about a natural generalization of the \(\mathsf {D}\) -cohomology of monoids to a cohomology theory for small categories first given by Charles Wells in an unpublished 1980 article. It is essentially the cohomology published later by Baues and Wirsching in 1985. The concluding seventh section gives proofs that \(\mathsf {D}\) -cohomology is equivalent, modulo a dimension shift, to relevant cases of Grothendieck sheaf cohomology and Beck cotriple cohomology, the letter being a theorem of Wells.