Global Actions

TL;DR

This paper develops a homotopy theory for global actions, algebraic structures analogous to topological spaces, and establishes a Galois correspondence between their coverings and fundamental groups.

Contribution

It introduces the homotopy theory of global actions and proves a Galois correspondence linking coverings to subgroups of the fundamental group.

Findings

Established a Galois correspondence for global actions

Connected coverings correspond to subgroups of the fundamental group

Developed foundational homotopy theory for global actions

Abstract

A global action is an algebraic analogue of a topological space. It consists of group actions $G_\alpha\curvearrowright X_\alpha$, $(\alpha\in\Phi)$, which fulfill a certain compatibility condition. We investigate the homotopy theory of global actions. The main result establishes a Galois type correspondence between connected coverings of a given connected global action and subgroups of the fundamental group of that action.