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

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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.
DOI : 10.4171/dm/573
Classification : 16D90, 16T05, 55U35
Mots-clés : Lie groupoids, Morita equivalence, bicategories, Hopf algebroids, weak equivalences, principal bundles, categorical groups, orbit spaces 
Laiachi El Kaoutit; Niels Kowalzig. Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences. Documenta mathematica, Tome 22 (2017), pp. 551-609. doi: 10.4171/dm/573
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     author = {Laiachi El Kaoutit and Niels Kowalzig},
     title = {Morita {Theory} for {Hopf} {Algebroids,} {Principal} {Bibundles,} and {Weak} {Equivalences}},
     journal = {Documenta mathematica},
     pages = {551--609},
     year = {2017},
     volume = {22},
     doi = {10.4171/dm/573},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/573/}
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