Monadic approach to Galois descent and cohomology

TL;DR

This paper presents a simplified, monadic categorical framework for Galois descent and cohomology, avoiding reliance on Grothendieck descent and providing explicit calculations for related algebraic structures.

Contribution

It introduces a monadic approach to Galois descent that clarifies the theory and connects it with categorical algebra and cohomology, with explicit computations.

Findings

Monadic approach simplifies Galois descent theory.

Explicit calculations with 1-cocycles and non-abelian cohomology.

Connections established between descent data and categorical algebra.

Abstract

We describe a simplified categorical approach to Galois descent theory. It is well known that Galois descent is a special case of Grothendieck descent, and that under mild additional conditions the category of Grothendieck descent data coincides with the Eilenberg-Moore category of algebras over a suitable monad. This also suggests using monads directly, and our monadic approach to Galois descent makes no reference to Grothendieck descent theory at all. In order to make Galois descent constructions perfectly clear, we also describe their connections with some other related constructions of categorical algebra, and make various explicit calculations, especially with 1-cocycles and 1-dimensional non-abelian cohomology, usually omitted in the literature.