Janelidze's Galois theory

Question

I am interested in learning about categorical Galois theory, as developed by Janelidze. I am a graduate student who has good familiarity with category theory, but not in the level of doing research on it.

First of all, I would like to ask where is the best place to study this theory in the greatest generality developed until now? Also, I know there are many papers, in different directions, dealing with Galois theory in category and topos theory - is Janelidze's theory "included" in some other theory, in the way Grothendieck's Galois theory is "included" in Janelidze's?

I have started reading the book Galois Theories. In section 1 of Chapter 5, the section that presents the theorem, Borceux and Janelidze use definitions that I haven't seen before, like "admissible class of morphisms" and "relatively admissible adjunction". Are these other concepts in disguise? And if they are notations created by the authors, do they have actual use anywhere else except from the development of the theory? I am asking all these questions because I am not sure what I should learn first before moving on to this generalization of Galois theory. I have tried reading other papers by Janelidze on this topic, e.g Pure Galois theory in Categories but the notation there seems very technical and the theorems are not proved.

Thanks for any help.

Answer 1

I only began to understand Galois Theory when I picked up Janelidzes book.

In the first chapter he covers the usual Galois Theory and mentions of course that an adjunction between posets is a Galois Connection. This is categorified up a categorical level as described here in nlab. They mention:

It is the particular case where the 2-relation $R$ is the hom-functor $S^{op}×S\rightarrow Set$; the corresponding adjunction is something which Lawvere calls conjugation.

In the second chapter, Janelidze covers the Galois Theory of Grothendieck, but not in:

in its full generality, that is in the context of Schemes: this would require a long technical introduction. But the spirit of Grothendiecks approach is applied to the context of fields.

These notes by Lenstra do. He also mentions Galois Categories (as Janelidze does not)and this links up with the Tannaka-Krein Reconstruction.

In the fourth chapter, besides mentioning the Pierce Spectrum, which is an interesting variation on the Zariski Spectrum and links up to Stone Spaces, he mentions effective monadic descent

In its more specific meaning descent is the study of generalizations of the sheaf condition on presheaves to presheaves with values in higher categories.

The descent is monadic when the pseudo-functor, aka the 2-presheaf, in its fibered category avatar (by the Grothendieck construction) are bifibrations. This allows the use of monads. He also mentions internal presheafs which are a concept in Enriched Category Theory.

In Chapter five, Janelidze relativises these concepts in the context of slice categories in preparation of proving the Abstract Categorical Galois Theorem.

This is extended in Chapter seven to 'Non-Galoisian Galois Theorem' by removing the Galois condition on effective descent, and by way of the Joyal-Tierney Theorem places it in the context of Grothendieck Toposes.

NLab goes on to say:

For $K$ a field let $Et(K)$ be its small étale site. And let $\bar{E}:=Sh(Et(K))$ be the sheaf topos over it. This topos is a

1. local topos
2. locally connected topos
3. connected topos

Then Galois extensions of $K$ correspond precisely to the locally constant objects in $\bar{E}$. The full subcategory on them is the Galois topos $Gal(\bar{E})\rightarrow \bar{E}$.

The Galois group is the fundamental group of the topos.

Hence, they conclude:

Accordingly in topos theory Galois theory is generally about the classification of locally constant sheaves. The Galois group corresponds to the fundamental group of the topos.

This can then be established in higher Topos Theory where a cohesive structure on the higher topos is used to make the constructions go through.

Proposition. For $H$, a ∞-topos that is:

1. locally ∞-connected
2. ∞-connected,

We have a natural equivalence $LConst(X)\backsimeq ∞Grpd[Π(X),∞Grpd_{\kappa}]$ of locally constant ∞-stacks on $X$ with ∞-permutation representations of the fundamental ∞-groupoid of $X$.

One, ought to here make the connection with the usual classification of topological covering spaces:

Let $X$ be a topological space, that is:

1. locally connected

2. connected

then, $Cov(X)\backsimeq Top[\pi (X),Set]$