Motivated by constructions in the theory of inverse semigroups and etale 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.
Keywords: Topos theory, inverse semigroups, \'etale groupoids, isotropy groups, crossed modules
2010 MSC: 18B25, 18B40, 18D05, 20L05, 20M18, 20M35, 22A22
Theory and Applications of Categories, Vol. 26, 2012, No. 24, pp 660-709.
Published 2012-11-16.
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