Galois theory in bicategories

TL;DR

This paper develops a Galois descent theory within bicategories, generalizing classical theorems and providing numerous examples including Hopf-Galois and Morita-Takeuchi theories, and introduces new comatrix corings.

Contribution

It extends Galois theory to bicategories, generalizing key theorems and introducing new structures like comatrix corings based on quasi bialgebras.

Findings

Generalized Beck's and Joyal-Tierney theorems

Unified framework for various Galois theories

Construction of new comatrix corings

Abstract

We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras.