Isotropy and Crossed Toposes

Motivated by constructions in the theory of inverse semigroups and tale groupoids, we define and investigate the concept of isotropy from a topos-theoretic perspective.Our main conceptual tool is a monad on the category of grouped toposes.Its algebras correspond to a generalized notion of crossed module, which we call a crossed topos.As an application, we present a topos-theoretic characterization and generalization of the 'Clifford, fundamental' sequence associated with an inverse semigroup.