This chapter is the core of the book. It could even be given a longer title, namely Categorical Galois theorem, “non-Grothendieck” examples and factorization systems.

We show first how the situation for commutative rings, studied in chapter 4, generalizes to develop Galois theory with respect to an axiomatic categorical setting. This setting consists basically in an adjunction with “well-behaved” properties, which mimic the situation of the Pierce spectrum functor and its adjoint, as in section 4.3. This categorical setting will contain the situations of the previous chapters: in particular the cases of fields and commutative rings. But our categorical Galois theorem will also apply to many other contexts.

First we apply the categorical Galois theorem to the study of central extensions of groups. This topic is generally not considered as part of Galois theory, but neverthesless the central extensions of groups turn out to be precisely the objects split over extension in a special case of categorical Galois theory.

This chapter also provides a good help for understanding the relationship between the Galois theory and factorization systems, which began in [18]. We focus in particular on the case of semi-left-exact reflections and apply it to the monotone–light factorization of continuous maps between compact Hausdorff spaces. It should be noticed that the situation of chapter 4, the categorical Galois theory of rings, is also a special case of a semi-left-exact reflection.

It might look strange for a topologist to describe light maps of compact Hausdorf spaces as the actions of a compact totally disconnected equivalence relation considered as a topological groupoid.