Topos Theoretic Aspects of Semigroup Actions

We define the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classified by the topos associated with an inverse semigroup. Unlike in the group case, not all set-theoretic torsors are isomorphic: we shall give a complete description of the category of torsors. We explain how a semigroup prehomomorphism gives rise to an adjunction between a restrictions-of-scalars functor and a tensor product functor, which we relate to the theory of covering spaces and E-unitary semigroups. We also interpret for semigroups the Lawvere-product of a sheaf and distribution, and finally, we indicate how the theory might be extended to general semigroups, by defining a notion of torsor and a classifying topos for those.