  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
${\mathsf{Sh}}(X)$.
