The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra. For the topos of sets, we show that torsion-free functors on Loganathan's category $L(S)$ of an inverse semigroup $S$ are equivalent to a special class of non-strict representations of $S$, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of $S$. We describe the correspondence between directed and pullback preserving functors on $L(S)$ and transitive and effective representations of $S$, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation of an inverse semigroup in the topos of sheaves ${\mathsf{Sh}}(X)$ on a topological space $X$ as well as outline an approach on how to define such a representation in an arbitrary topos. We prove that the category of filtered functors from $L(S)$ to the topos ${\mathsf{Sh}}(X)$ is equivalent to the category of universal representations of $S$ in ${\mathsf{Sh}}(X)$.