Algebra of Principal Fibre Bundles, and Connections

TL;DR

This paper explores the algebraic structures underlying principal fibre bundles and their connections, emphasizing the role of groupoids as introduced by Charles Ehresmann to unify various aspects of fibre bundle theory.

Contribution

It synthesizes efforts to algebraize fibre bundle theory, highlighting the significance of groupoids and their actions in understanding connections.

Findings

Groupoids provide a unifying algebraic framework for fibre bundle theory.

Connections can be characterized using groupoid actions.

The algebraic perspective offers new insights into fibre bundle structures.

Abstract

We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.