Geodesic
Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences
Documenta mathematica, Tome 22 (2017), pp. 551-609.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 16T05, 16D90, 18D05, 18D10, 55U35
Keywords: Hopf algebroids, weak equivalences, Morita equivalence, principal bundles, bicategories, categorical groups, orbit spaces, Lie groupoids 
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     author = {El Kaoutit, Laiachi and Kowalzig, Niels},
     title = {Morita {Theory} for {Hopf} {Algebroids,} {Principal} {Bibundles,} and {Weak} {Equivalences}},
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El Kaoutit, Laiachi; Kowalzig, Niels. Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences. Documenta mathematica, Tome 22 (2017), pp. 551-609. http://geodesic.mathdoc.fr/item/DOCMA_2017__22__a36/