A 2-colimit characterization of internal categories of torsors

Abstract

The present work generalizes the classical result classifying isomorphism classes of locally trivial principal fiber bundles with an abelian structure group by means of equivalence classes of Čech cocycles. In the generalization the structure group is replaced by a category object in a Barr-exact category, the locally trivial principal fiber bundles are replaced by locally representable internal presheaves and the first cohomology group is replaced by the 2-dimensional direct limit of a suitable pseudofunctor into CAT.