Pregroupoids and their enveloping groupoids

TL;DR

This paper establishes a construction for associating an enveloping groupoid to any pregroupoid, demonstrating the equivalence of torsors and pregroupoids, and extending these results to principal fibre bundles.

Contribution

It introduces a left adjoint functor from pregroupoids to groupoids, providing a systematic way to obtain enveloping groupoids and linking torsors with pregroupoids.

Findings

Existence of a left adjoint functor from pregroupoids to groupoids

Equivalence between torsors and pregroupoids

Enveloping groupoids for principal fibre bundles

Abstract

We prove that the forgetful functor from groupoids to pregroupoids has a left adjoint, with the front adjunction injective. Thus we get an enveloping groupoid for any pregroupoid. We prove that the category of torsors is equivalent to that of pregroupoids. Hence we also get enveloping groupoids for torsors, and for principal fibre bundles.