On the cohomology of the inverse semigroup G of the G-sets of a groupoid G

Abstract

Renault has defined in [7] the cohomology of the inverse semigroup G of the G-sets of a given groupoid G as a functor from the category of G-presheaves to that of abelian groups. We show in our paper that G-presheaves is isomorphic to AbD(G) where D(G) is the division category defined from Loganathan in [6] and used there to give another description of the Lausch cohomology of inverse semigroups. This isomorphism allows us in turn to prove that Renault and Lausch cohomology groups of G are isomorphic.