Fibre bundles in general categories
This article indicates how one can effectively describe the notion of fiber bundle in any category with finite inverse limits. This description revolves around the notion of a pregroupoid structure on a map \(\pi\) : \(X\to B\). Pregroupoids arise from groupoids and also include frame bundles and the universal covering space of a given space. In order to have a good theory of fiber bundles, the notion of descent map is needed. A regular pregroupoid is given by \(\pi\) : \(X\to B\), where \(\pi\) and \(X\to 1\) are descent maps. Given this descent data, one can associate a group \(X_*\) and a groupoid \(X^*\) which act fiberwise on \(X\to B\) on the right and left, respectively. These actions lead to an equivalence of the category of regular pregroupoids over B with the category of regular principal homogeneous group actions \(X\times G\to X\) with X/G\(\simeq B\). Furthermore, the fiber bundles over X can be viewed in terms of either \(X_*\) or \(X^*\) actions and this relationship is analyzed in detail. The final section of the paper applies these considerations to the category of locales. It should be pointed out that the constructions in this paper have been refined and developed further by the author to allow for describing ``generalized'' fiber bundles [Lect. Notes 1348, 194-207 (1988; Zbl 0657.18007)]. These generalizations provide a framework for discussing foliations and their holonomy groupoids.
category of regular principal homogeneous group actions
scientific article; zbMATH DE number 4087892
scientific article