Ganna Kudryavtseva in Primož Škraba: Filtered functors and internal actions related to inverse semigroups, part 2

Abstract: This talk is a contribution to the development of the theory of internal actions of inverse semigroups in toposes. We prove that the category of filtered functors from the Loganathan’s category L(S) of an inverse semigroup S to the topos of sheaves Sh(X) on a topological space X is equivalent to the category of universal representations of S in Sh(X). This upgrades to Sh(X) the result proved by Funk and Hofstra for the topos of sets. For the topos of sets, we show that torsion-free functors on L(S) are equivalent to a special class of non-strict representations of S, which we call connected, whereas the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S.