On synthetic interpretation of quantum principal bundles

TL;DR

This paper reinterprets quantum principal bundles as principal bundles in Synthetic Noncommutative Differential Geometry, establishing a new categorical framework and linking it to Hopf-Galois extensions.

Contribution

It introduces a notion of noncommutative principal bundles within braided monoidal categories and connects it to faithfully flat Hopf-Galois extensions.

Findings

Noncommutative principal bundles are defined in braided monoidal categories.

A noncommutative principal bundle in the opposite category of vector spaces equals a faithfully flat Hopf-Galois extension.

Framework bridges quantum principal bundles with algebraic structures in noncommutative geometry.

Abstract

Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within a braided monoidal category is introduced and it is shown that a noncommutative principal bundle in the category opposite to the category of vector spaces is the same as a faithfully flat Hopf-Galois extension.