Principal bundles as Frobenius adjunctions with application to geometric morphisms

Using a suitable notion of principalG-bundle, defined relative to an arbitrary cartesian category, it is shown that principal bundles can be characterised as adjunctions that stably satisfy Frobenius reciprocity. The result extends from internal groups to internal groupoids. Since geometric morphisms can be described as certain adjunctions that are stably Frobenius, as an application it is proved that all geometric morphisms, from a localic topos to a bounded topos, can be characterised as principal bundles.

Monodromy groupoid of an internal groupoid in topological groups with operations

In this paper, the monodromy groupoids of internal groupoids in the topological groups with operations are studied and a monodromy principle for internal groupoids in groups with operations is obtained.