On the existence of bibundles

TL;DR

This paper investigates the conditions for the existence of bibundles, linking their existence to the outer automorphism group of G, and develops a comprehensive classification theory including examples related to loop group bundles.

Contribution

It introduces a general theory for bibundles associated with crossed-modules, connecting their existence to outer automorphisms and providing a classification framework.

Findings

Existence of bibundles relates to the structure of Out(G).

Developed a classifying theory for bibundles in full generality.

Examples demonstrate links with loop group bundles.

Abstract

We consider the existence of bibundles, in other words locally trivial principal $G$ spaces with commuting left and right $G$ actions. We show that their existence is closely related to the structure of the group $\Out(G)$ of outer automorphisms of $G$. We also develop a classifying theory for bibundles. The theory is developed in full generality for $(H, G)$ bibundles for a crossed-module $(H, G)$ and we show with examples the close links with loop group bundles.