Semigroups and Toposes

Abstract

We establish a semigroup presentation of the classifying topos of an inverse semigroup. For this purpose we consider a category larger than inverse semigroups and prehomomorphisms, whose objects we call left *-semigroups. We show that any etendue has a left *-semigroup presentation. We show that any left *-semigroup may be codensely embedded in an inverse semigroup. We also establish a strictly topos description of E-unitary inverse semigroups.