A 2-Site for Continuous 2-Group Actions

TL;DR

This paper generalizes Elmendorf's Theorem to the setting of topological 2-groups, establishing a 2-categorical equivalence with 2-sheaves on a 2-site, thus extending the framework of continuous group actions.

Contribution

It introduces a 2-dimensional generalization of Elmendorf's Theorem, connecting 2-group actions with 2-sheaves on a 2-site, expanding the categorical understanding of topological 2-groups.

Findings

Establishes a 2-equivalence between 2-group actions and 2-sheaves.

Extends the topos-theoretic framework to 2-categories.

Provides a foundation for further study of higher group actions.

Abstract

Elmendorf's Theorem states that the category of continuous actions of a topological group is a Grothendieck topos in the sense that it is equivalent to a category of sheaves on a site. This paper offers a 2-dimensional generalization by showing that a certain 2-category of continuous actions of a topological 2-group is 2-equivalent to a 2-category of 2-sheaves on a suitable 2-site.