Galois category

Floor 2, Les sutures · introduced in Chapter 5, Courte marche à travers la théorie de Galois

What are objects and seams, read in Grothendieck’s Galois theory?

A category of finite sets with an action, and a functor that forgets the action: the objects of a group are its connected objects, seams its isomorphisms, markings identifications of fibre functors, and seams over automorphisms its twists by Out(G).

Plate 2.7The Galois category of

S_3

, read in fields: the fields between

\Q

and

\Q(\sqrt[3]2,\omega)

over the subgroups that fix them. The three cubic fields, gold, are three incarnations of one rigid object, joined by exactly one

\Q

-isomorphism each; complex conjugation fixes

\sqrt[3]2

and is the automorphism of

\Q(\omega)

Definition

A category equivalent to the category of finite continuous $\pi$ -sets, for a profinite group $\pi$ , is a Galois category, and a functor $F$ to finite sets satisfying Grothendieck’s axioms is a fundamental functor, or fibre functor. For a finite group $G$ , the category $\mathcal C_G$ of finite $G$ -sets and $G$ -maps, with the functor $F_G$ that forgets the action, is one, and $\Aut(F_G)\cong G$ .

Theorem(Objects are the connected objects of a Galois category) proved

Let $\mathcal C_G$ be the category of finite $G$ -sets and $G$ -maps, and $F_G\colon\mathcal C_G\to\mathbf{FinSet}$ the functor that forgets the action.

(a) $(\mathcal C_G,F_G)$ satisfies Grothendieck’s axioms (G1)–(G6), and $g\mapsto(x\mapsto gx)$ is an isomorphism from $G$ onto $\Aut(F_G)$ .

(b) The connected objects of $\mathcal C_G$ are the objects of $G$ , its transitive sets. The object $G/H$ is Galois, that is, its automorphism group acts transitively on its fibre, if and only if $H$ is normal.

(c) A Galois category whose fundamental functor has automorphism group isomorphic to $G$ is equivalent to $\mathcal C_G$ , by an equivalence that carries the fundamental functor to $F_G$ .

Proof

(a) Limits, sums, and quotients by finite groups of $G$ -automorphisms are formed on the underlying sets. The image of a $G$ -map and its complement are $G$ -stable, so a $G$ -map is a surjection onto its image followed by the inclusion of a direct summand, and a bijective $G$ -map is a $G$ -isomorphism: $F_G$ is exact and reflects isomorphisms. Let $\eta$ be an automorphism of $F_G$ and put $g=\eta_G(1)$ , its value at the regular set. For $x\in X$ the map $f_x(k)=kx$ is a $G$ -map, and naturality gives $\eta_X(x)=\eta_X(f_x(1))=f_x(\eta_G(1))=gx$ . Conversely each $g$ defines an automorphism of $F_G$ , and composition corresponds to multiplication.

(b) A $G$ -set is the sum of its orbits, so it is connected exactly when it is transitive. The automorphism group of $G/H$ is $N_G(H)/H$ , acting freely on $G/H$ , and transitively exactly when $N_G(H)=G$ . (c) is Grothendieck’s theorem (SGA 1, Exposé V).

Proposition(Markings are fibre-functor data) proved

Let a theory supply a finite group $\Gamma$ acting on a set $Y$ , and let $\mu\colon G\to\Gamma$ be a marking.

(a) Restriction along $\mu$ is an exact functor $\mu^*\colon\mathcal C_\Gamma\to\mathcal C_G$ with $F_G\circ\mu^*=F_\Gamma$ , and the marked set is $Y_\mu=\mu^*(Y)$ .

(b) Conversely, let $H\colon\mathcal C_\Gamma\to\mathcal C_G$ be a functor and $\eta\colon F_G\circ H\to F_\Gamma$ an isomorphism of functors. There is exactly one homomorphism $\mu\colon G\to\Gamma$ for which every $\eta_Y$ is a $G$ -isomorphism $H(Y)\to Y_\mu$ . Replacing $\eta$ by $\gamma\circ\eta$ , for $\gamma\in\Gamma=\Aut(F_\Gamma)$ , replaces $\mu$ by $\gamma\mu(\cdot)\gamma^{-1}$ .

(c) In particular, a change of marking by an inner automorphism is a change of the identification of fibre functors, that is, of base point: the maps $y\mapsto\mu(h)y$ form an isomorphism of functors $\mu^*\to(\mu\circ\iota_h)^*$ , where $\iota_h(g)=hgh^{-1}$ .

Proof

(a) is immediate. (b) For $g\in G$ , the maps $x\mapsto gx$ on the sets $F_G(H(Y))$ form an automorphism of $F_G\circ H$ . Transported by $\eta$ it is an automorphism of $F_\Gamma$ , hence the action of a unique element $\mu(g)\in\Gamma$ . Then $\eta_Y(gx)=\mu(g)\eta_Y(x)$ for all $Y$ and $x$ , and $\mu$ is a homomorphism; it is the only one with this property, because $\Gamma$ acts faithfully on its regular set. Replacing $\eta$ by $\gamma\eta$ conjugates the transported automorphisms by $\gamma$ . (c) $\mu(h)\mu(g)y=\mu(hgh^{-1})\mu(h)y$ .

Theorem(Seam monodromy is non-abelian

H^{1}

) proved

Let $\mathcal G$ be a connected graph with base vertex $v$ and $\pi=\pi_1(\mathcal G,v)$ , and let $X$ be an object with stabilizer $H$ and automorphism group $A=\Aut_G(X)\cong N_G(H)/H$ .

(a) The gauge classes of seam systems for $X$ over $\mathcal G$ correspond to $H^1(\pi,A)=\operatorname{Hom}(\pi,A)/A$ , the first non-abelian cohomology set for the trivial action. They also correspond to the isomorphism classes of principal $A$ -coverings of $\mathcal G$ : to a system corresponds the covering whose fibre over a vertex $w$ is the set of alignments $\Iso_G(X,Y_w)$ , with $A$ acting by precomposition and the seams as transition maps. The system is coherent exactly when its class is trivial.

(b) Let $\Aut^{\mathrm{tw}}(X)$ be the group of pairs $(\beta,f)$ , with $\beta\in\Aut(G)$ and $f$ a seam over $\beta$ from $X$ to $X$ , an extension $1\to A\to\Aut^{\mathrm{tw}}(X)\to\Aut(G)_{[H]}\to1$ . With an incarnation of $X$ at each vertex, a seam over an automorphism on each edge, and changes of marking as gauge transformations, such systems correspond to $H^1(\pi,\Aut^{\mathrm{tw}}(X))$ . Their image in $H^1(\pi,\Aut(G)_{[H]})$ is the monodromy of the markings, and since $\pi$ is free, every class there arises.

Proof

(a) The correspondence with $\operatorname{Hom}(\pi,A)/A$ is the gauge theorem of Chapter 4. A principal $A$ -covering of a connected graph is determined up to isomorphism by its holonomy, a homomorphism $\pi\to A$ up to conjugation, and in a gauge the transition maps of the covering of alignments are the link variables. (b) The gauge theorem uses only that link variables compose in a group, so it applies with $\Aut^{\mathrm{tw}}(X)$ in place of $A$ . A homomorphism from a free group lifts along a surjection, generator by generator.

Example(The cube roots of two)

Let $K=\Q(\sqrt[3]2,\omega)$ , $\omega=e^{2\pi i/3}$ , with group $G\cong S_3$ . The fields $\Q(\sqrt[3]2)$ , $\Q(\omega\sqrt[3]2)$ and $\Q(\omega^2\sqrt[3]2)$ are the fixed fields of the three subgroups of order 2. They are three incarnations of the object $S_3/C_2$ , which is rigid because $C_2$ is self-normalizing. So between two of them there is exactly one $\Q$ -isomorphism, such as $\sqrt[3]2\mapsto\omega\sqrt[3]2$ , and these compose coherently. The field $\Q(\omega)=K^{C_3}$ has automorphism group $N(C_3)/C_3\cong C_2$ , complex conjugation. The description $\Q(\omega)\subset K$ forgets the three cube roots of 2, which form the fibre $C_3$ of $S_3\to S_3/C_3$ .

Remark(What the language of seams adds)

Everything in the Galois-categorical reading is classical: Galois categories, the Galois theory of fields in this form, categorical Galois theory, non-abelian $H^1$ and twisted forms, and gerbes. What the language of seams adds is an emphasis, not a theorem. Several theories meet one connected object through different pointed Galois categories $(\mathcal C_\Gamma,F_\Gamma)$ , joined by markings, and the subject is the isomorphisms that those theories single out and whether they cohere. A Galois category contains every isomorphism between two incarnations and prefers none. Which ones a construction picks out, and whether the picks commute around a cycle, are data that the Galois category does not contain. Likewise the statuses of a bridge record knowledge, not structure.

Built from: objectseammarkingseam over an automorphismseam monodromy
Objects: the object of size 168the object of size 1the object of size 56
In the Esquisse: 16Une loi de réciprocité17L’écart de GaloisÉp.L’horizon : dessins d’enfants