Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences

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.

2010 Mathematics Subject Classification: Primary 16D90, 16T15, 18D05, 18D10; secondary 14M17, 22A22, 58H05

Keywords and Phrases: Hopf algebroids, weak equivalences, Morita equivalence, principal bundles, bicategories, categorical groups, orbit spaces, Lie groupoids.

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