JournalsdmVol. 22pp. 551–609

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

  • Laiachi El Kaoutit

    Universidad de Granada, Departamento de Álgebra and IEMath-Granada, Facultad de Educación Economía y Tecnología, Campus De Ceuta Cortadura del Valle s/n., 51001 Ceuta, España

  • Niels Kowalzig

    Dipartimento di Matematica, Università degli Studi di Roma La Sapienza, P.le Aldo Moro 5, 00185 Roma, Italia
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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.

Cite this article

Laiachi El Kaoutit, Niels Kowalzig, Morita Theory for Hopf Algebroids, Principal Bibundles, and Weak Equivalences. Doc. Math. 22 (2017), pp. 551–609

DOI 10.4171/DM/573