Pullback of quantum principal bundles

TL;DR

This paper develops an abstract framework to extend the classical notion of pullback to compact quantum principal bundles, enriching the theory of equivariant topology with categorical and quantum insights.

Contribution

It introduces a new abstract notion of Cartesian squares applicable beyond fiber products, enabling the extension of pullback concepts to quantum principal bundles and related structures.

Findings

Extended pullback concept to quantum principal bundles

Embedded equivariant topology into Grothendieck categories

Unified classical and quantum principal bundle theory

Abstract

We introduce an abstract framework of Cartesian squares beyond the context of fiber products, and use it to extend the notion of pullback from classical to compact quantum principal bundles. Based only on our abstract notion of a Cartesian square, we extend key concepts of Equivariant Topology, such as the pullback of a family of group actions, orbit spaces, slices and global sections, change of base and structure group, free actions, and the groupoid of compact principal bundles. Finally, we embed the thus extended Equivariant Topology inside the 2-category of Grothendieck categories in such a way that our notion of a Cartesian square becomes the appropriate Beck-Chevalley condition.