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Grothendieck's Galois theory

Idea

For a sufficiently nice topological space, the fundamental group at a point can be reconstructed as a group of deck transformations of the universal covering space, which is the same as the automorphisms of the fiber over that point of the projection map. The deck transformations are monodromies induced by loops at the base point; intuitively they can be taken infinitesimally, and viewed as symmetries of the point. The functor which assigns to a point the fiber functor over it, generalizes to fiber functors in Tannakian formalism of Grothendieck which defines in more general setups the fundamental groupoid as the group of automorphisms of the appropriate fiber functor. See also fundamental group of a topos.

Grothendieck’s Galois theory was constructed in order to define for schemes an analogue of the familiar correspondence

covering spaces of $X$ : ${\pi }_1(X)$-sets

for a locally path connected, semilocally simply connected topological space $X$.

The objects on the left are not difficult to define for schemes (at least naively – one really needs trivialisations over étale covers), but it may not be entirely immediate what the fundamental group defined in terms of loops should be.

The reason Galois’s name is attached to this theory is that in the case of the scheme $\mathrm{Spec}(k)$, the objects corresponding to the covering spaces are simply field extension?s $\mathrm{Spec}(k\prime )$. The fundamental group of schemes defined in this way is the algebraic fundamental group, and is a profinite group.

Generalizations

The basic idea of Grothendieck’s Galois theory may be extended to objects in an topos – leading to a notion of fundamental group of a topos – and then further to objects in any (∞,1)-topos. For more on this see homotopy group of an ∞-stack.

Details

Preliminaries

Definition

Given an arrow $f:x\to y$ in a category $C$ the category of arrows compatible with $f$, denoted $\mathrm{Comp}(f)$ is the full subcategory of the undercategory $x\downarrow C$ on the arrows that coequalize the same pairs $g,h:w\rightrightarrows x$ that $f$ does.

Definition

An arrow $f:x\to y$ in a category $C$ is a strict epimorphism if it is initial in $\mathrm{Comp}(f)$.

It is not obvious, but a strict epimorphism is an epimorphism.

Grothendieck’s axioms

In what follows, Let $C$ be a category and $F:C\to \mathrm{Set}$ a functor. The axioms presented here are as in

• Eduardo Dubuc, C. S. de la Vega On the Galois theory on Grothendieck, Bol. Acad. Nac. Cienc. (Cordoba) 65 (2000) 111–136. arXiv

Some terminology: $X\in C$ is called finite if $F(X)$ is a finite set. Let ${\int }_{F}C$ denote the category of elements of $F$, in which an object $(X,a)$ is called finite if $X$ is finite.

• G0) The full subcategory of ${\int }_{F}C$ on the finite objects is cofinal?.

• G1) $C$ has all finite limits

• G2) $C$ has an initial object, finite coproducts and quotients by finite groups

• G3) Given $f:x\to z$ in $C$ there is a factorisation $x\stackrel{e}{\to }y\stackrel{i}{\to }z$ where $e$ is a strict epimorphism and $i$ is a mono. Also, $y$ is assumed to be a direct summand of $z$.

• G4) $F$ preserves finite limits

• G5) $F$ preserves initial object, finite sums, quotients by finite group actions and sends strict epimorphisms to surjections

• G6) $F$ reflects isomorphisms

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It follows from the axioms that $F$ is a pro-representable functor?. The automorphism group of the pro-object $P$ representing $F$ is (should be. I’m not familiar enough with pro-objects) a profinite group $\pi$. This acts on $F(X)=[P,-]$ by precomposition (talking out of my depth here – it’s getting a bit vague) and so $F$ lifts to a functor to $\pi -\mathrm{Set}$, and Grothendieck’s result is that this functor is an equivalence of categories.

There are several modifications one can make the above. In the case that $C$ is the category of covering spaces of a nice enough space, the functor $F$ is representable by the universal covering space, and so there is a ‘representable’ version of the above, not needing to utilise profinite groups. One can also consider just the connected-objects version, and end up with an equivalence to the category of transitive $\pi$-sets.

The classical case of fields

Even for the classical case of the inclusion of fields, Grothendieck’s Galois theorem gives more general statement than the previously known. This is the Grothendieck’s version of the Galois correspondence theorem for fields:

Let $K\subset L$ be a finite dimensional Galois extension of fields. Let $\mathrm{Gal}[L:K]$ denote the group of $K$-automorphisms of $L$ and $\mathrm{Gal}[L:K]-\mathrm{finSet}$ the category of finite $\mathrm{Gal}[L:K]$-sets. By ${\mathrm{SplitfinAlg}}_K(L)$ denote the finite dimensional $K$-algebras which are split over $L$; here $L$ itself is an object. Consider the representable functor $h_L={\mathrm{Hom}}_K(-,L):{\mathrm{SplitfinAlg}}_K(L)\to \mathrm{Set}$. It takes values in the subcategory of finite sets and it comes with a canonical $\mathrm{Gal}[L:K]$-action. In other words, this functor factors through $\mathrm{Gal}[L:K]-\mathrm{finSet}$. Moreover, the corresponding functor

$${\mathrm{SplitfinAlg}}_K(L)\to \mathrm{Gal}[L:K]-\mathrm{finSet}$$SplitfinAlg_K(L)\to Gal [L : K]-finSet

is an equivalence of categories.

There is an infinitary version as well, generalizing the classical Galois theorem on infinitary Galois extensions.

Thus let $K\subset L$ be an arbitrary Galois extension. Now $\mathrm{Gal}[L:K]$ denotes the profinite Galois group and $\mathrm{Gal}[L:K]-\mathrm{profinSpace}$ the category or profinite $\mathrm{Gal}[L:K]$-spaces. ${\mathrm{SplitAlg}}_K(L)$ denotes the category of $K$-algebras split over $L$ (possible infinite-dimensional). Then there is a canonical antiequivalence of categories

$${\mathrm{SplitAlg}}_K(L)\to \mathrm{Gal}[L:K]-\mathrm{profinSpace}$$SplitAlg_K(L)\to Gal [L : K]-profinSpace

(factorizing a profinite-space version of the representable functor ${\mathrm{Hom}}_K(-,L)$).

Galois theorem for locales and topoi

Let $E$ be a Grothendieck topos. Then there exist an open localic groupoid $G$ such that $E$ is equivalent to the category of étale presheaves over $G$. One of the classical references is

• J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Inst. of Fund. Res. Lectures on Mathematics 40, Bombay, 1967. iv+176+iv pp.

This is a variant of the theorem in the setting of locales from

• Andre Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

References

The original development of the theory by Grothendieck is in SGA1.

A more recent treatment can be found in

• Eduardo Dubuc and C. S. de la Vega On the Galois theory on Grothendieck, Bol. Acad. Nac. Cienc. (Cordoba) 65 (2000) 111–136. arXiv

Basic intuition is explained in

• Pierre Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, pdf

The construction for general toposes is described in section 8.4 of

• Peter Johnstone, Topos theory , Academic Press (1977)

and, a current state of the art decription is in

• Marta Bunge, Galois groupoids and covering morphisms in topos theory, Galois theory, Hopf algebras, and semiabelian categories, 131–161, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, links.

A modern approach from classical via Grothendieck up to categorical Galois theory based on precategories and adjunctions is in

• F. Borceux, G. Janelidze, Galois theories, Cambridge Studies in Adv. Math. 72, 2001. xiv+341 pp.

The application of a general Tannakian theorem of Saavaedra Rivano, as corrected by Deligne, to the “differential Galois theory” for differential instead of algebraic equation is in the last chapter of Deligne’s Catégories Tannakiennes.