Monadic approach to Galois descent and cohomology

Francis Borceux, Stefaan Caenepeel and George Janelidze

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.

Keywords: Descent theory, Galois theory, monadic functor, group cohomology

2000 MSC: 18C15, 18D10

Theory and Applications of Categories, Vol. 23, 2010, No. 5, pp 92-112.

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