Abstract: } The topos $B(S)$ of an inverse semigroup $S$ is the category of `\'etale $S$-sets' $X\rightarrow E$, where E=E(S)$=idempotents of$S$. By definition, an isotropy torsor of an inverse semigroup$S$is a globally supported \'etale$S$-set for which the universal action by the isotropy group$Z(E)\rightarrow E$is free and transitive ($Z(E)$=idempotent centraliser). I will explain that there is a bijective correspondence between:

\begin{enumerate}

\item[(i)] isomorphism classes of isotropy torsors;

\item[(ii)] isomorphism classes of \'etale sections of the isotropy quotient of $B(S)$ in the sense of geometric morphisms;

\item[(iii)] central isomorphism classes of homomorphic sections of the maximum idempotent-separating congruence on $S$,

where (let us say) two such sections are centrally isomorphic if one is the conjugate of the other by an order preserving map $t:E\rightarrow Z(E)$

such that for all idempotents $e$, $t(e) ^*t(e)=e$.

\end{enumerate}

This explanation illustrates in part not only the natural relationship between inverse semigroups and toposes,

but also some aspects of isotropy theory for toposes in general.

[Joint work with Pieter Hofstra and Benjamin Steinberg.]