Morita theory for Hopf algebroids, principal bibundles, and weak equivalences

TL;DR

This paper establishes a Morita theory framework for flat commutative Hopf algebroids, linking weak equivalences, principal bibundles, and bicategory structures, thus providing an algebraic analogue to Lie groupoid Morita theory.

Contribution

It proves that Morita equivalence of Hopf algebroids is characterized by weak equivalences and principal bibundles, and constructs a universal bicategory framework for these structures.

Findings

Morita equivalence characterized by weak equivalences and principal bibundles

Existence of a bicategory of principal bundles with a universal 2-functor

Positive resolution of a conjecture by Hovey and Strickland

Abstract

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.