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Galois theory, a logical path from Grothendieck’s version to the fundamental theorem

Johan Felipe Garc´ ıa Vargas

Departamento de Matem´ aticas Universidad de los Andes

July 11, 2017

Outline

Traditionally Galois theory is seen as a correspondence given by

stabilizers and fixed points, Grothendieck frames it as a monadicity result.

In model theory, internality implies pro-definable binding group and

Galois correspondence. I will explain this from a categorical logic perspective as a natural consequence of a monadicity result.

In this framework, Grothendieck’s Galois theory is internality over

finite sets and Tannakian duality can be immersed as internality

• ver constructible sets.

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Outline

Traditionally Galois theory is seen as a correspondence given by

stabilizers and fixed points, Grothendieck frames it as a monadicity result.

In model theory, internality implies pro-definable binding group and

Galois correspondence. I will explain this from a categorical logic perspective as a natural consequence of a monadicity result.

In this framework, Grothendieck’s Galois theory is internality over

finite sets and Tannakian duality can be immersed as internality

• ver constructible sets.

2 of 8

Outline

Traditionally Galois theory is seen as a correspondence given by

stabilizers and fixed points, Grothendieck frames it as a monadicity result.

In model theory, internality implies pro-definable binding group and

Galois correspondence. I will explain this from a categorical logic perspective as a natural consequence of a monadicity result.

In this framework, Grothendieck’s Galois theory is internality over

finite sets and Tannakian duality can be immersed as internality

• ver constructible sets.

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Galois theory of fields

Fundamental theorem

Let F ≤ E be a Galois extension of fields, then the Galois group G = Aut(E/F) is pro-finite and there is a biyective correspondence between intermediate fields and closed subgroups of G. {K | F ≤ K ≤ E} {H ≤ G | H is closed}

Aut(E/K) E H

Grothendieck’s version

Let ω : C → Setsf be a fundamental functor from a Galoisian category, then π = Aut(ω) is a pro-finite group and ω lifts to an equivalence between C and the category of finite continuous π-actions Setsπ.

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Galois theory of fields

Fundamental theorem

Let F ≤ E be a Galois extension of fields, then the Galois group G = Aut(E/F) is pro-finite and there is a biyective correspondence between intermediate fields and closed subgroups of G.

Grothendieck’s version

Let ω : C → Setsf be a fundamental functor from a Galoisian category, then π = Aut(ω) is a pro-finite group and ω lifts to an equivalence between C and the category of finite continuous π-actions Setsπ

f .

C Setsπ

f

Setsf

ω

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Galois theory of fields

Fundamental theorem

Let F ≤ E be a Galois extension of fields, then the Galois group G = Aut(E/F) is pro-finite and there is a biyective correspondence between intermediate fields and closed subgroups of G.

Grothendieck’s version

Let ω : C → Setsf be a fundamental functor from a Galoisian category, then π = Aut(ω) is a pro-finite group and ω lifts to an equivalence between C and the category of finite continuous π-actions Setsπ

f .

In particular, let Cop be the category of finite ´ etale F-algebras split by E and take ω(X) = X(E) = HomF–alg.(A, E) when X = Spec A in C, then Aut(ω) = Aut(E/F) and C is equivalent to SetsG

f .

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Categorical logic

Given a first order theory T

The category of T-definables

(including imaginary sorts) T = Def(T eq) is a boolean pre-topos.

Models are logical functors

M : T → Sets.

Interpretations are logical

functors ι : T0 → T .

ι is an immersion if

ιX : SubT0(X) → SubT (ιX) is an isomorphism for every X.

× µ K V V Function µ : K × V → V .

b b

Constant 0 ∈ V < × Q Q A binary relation < in the sort Q.

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Categorical logic

Given a first order theory T

The category of T-definables

(including imaginary sorts) T = Def(T eq) is a boolean pre-topos.

Models are logical functors

M : T → Sets.

Interpretations are logical

functors ι : T0 → T .

ι is an immersion if

ιX : SubT0(X) → SubT (ιX) is an isomorphism for every X.

× µ K V V Function µ : K × V → V .

b b

Constant 0 ∈ V

< × Q Q A binary relation < in the sort Q.

4 of 8

Categorical logic

Given a first order theory T

The category of T-definables

(including imaginary sorts) T = Def(T eq) is a boolean pre-topos.

Models are logical functors

M : T → Sets.

Interpretations are logical

functors ι : T0 → T .

ι is an immersion if

ιX : SubT0(X) → SubT (ιX) is an isomorphism for every X.

Def(T0) Def(T ) × µ K V V

Every M | = T induces a M0 | = T0.

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Categorical logic

Given a first order theory T

The category of T-definables

(including imaginary sorts) T = Def(T eq) is a boolean pre-topos.

Models are logical functors

M : T → Sets.

Interpretations are logical

functors ι : T0 → T .

ι is an immersion if

ιX : SubT0(X) → SubT (ιX) is an isomorphism for every X.

Def(T0) Def(T ) × µ K V V

Every M | = T induces a M0 | = T0.

4 of 8

Categorical logic

Given a first order theory T

The category of T-definables

(including imaginary sorts) T = Def(T eq) is a boolean pre-topos.

Models are logical functors

M : T → Sets.

Interpretations are logical

functors ι : T0 → T .

ι is an immersion if

ιX : SubT0(X) → SubT (ιX) is an isomorphism for every X.

Def(T0) Def(T ) × µ K V V

Every M | = T induces a M0 | = T0. In fact, ι∗ : Mod(T) → Mod(T0) is an equivalence, if and only if, ι is an equivalence.

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A) T T A = Def(T, A) Sets

A A=dcl(A) ΓA=Hom(1,?)

Definition

ι is an internal cover if it’s stable and every T-definable is T0-internal.

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A) T T A = Def(T, A) Sets

A A=dcl(A) ΓA=Hom(1,?)

T0 T A0 T T A

A0 ι ιA A

Definition

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A)

Def(T0) Def(T ) X Z

b

b Bb

Definition

ι is an internal cover if it’s stable and every T-definable is T0-internal.

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A)

Def(T0) Def(T ) X Z Z′

b

b Bb

b a

Aa

Definition

ι is an internal cover if it’s stable and every T-definable is T0-internal.

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A)

Def(T0) Def(T ) Y Xi Xj · · ·

b b

Definition

ι is an internal cover if it’s stable and every T-definable is T0-internal.

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Definable closure and internal covers

Given A ⊆ M |

= T, the functor A = dcl(A) preserves limits and co-limits, but not necessarily images (i.e. ∃ quantifier).

ι : T0 → T is a stable

immersion if ιA is an immersion for every A.

Y is T0-internal over A if for

every M | = T A, M(Y ) = dcl(M0 ∪ A)

Def(T0) Def(T ) Y Xi Xj · · ·

b b

Definition

ι is an internal cover if it’s stable and every T-definable is T0-internal.

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Monadicity and definability of the binding group

Lemma (Kamensky)

ι : T0 → T is a stable immersion iff Ind ι : Ind T0 → Ind T is a cartesian closed functor.

Proof.

Use that 2 = 1 + 1 is the subobject classifier.

Lemma

If A : T0 → Sets contains a basis for the internal cover ι (i.e. every Y is T0-internal over A), then ιA is an equivalence.

Lemma

If T is a complete theory, for every A the functor Pro A : Pro T → Pro T A is monadic.

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Monadicity and definability of the binding group

Lemma (Kamensky)

ι : T0 → T is a stable immersion iff Ind ι : Ind T0 → Ind T is a cartesian closed functor.

Lemma

If A : T0 → Sets contains a basis for the internal cover ι (i.e. every Y is T0-internal over A), then ιA is an equivalence.

Proof.

Use compactness (and co-products) to get a regular epi fa : ιX → Y , afterwards use stability (and effectiveness) to define φa : ι(X/Eb) → Y .

Lemma

If T is a complete theory, for every A the functor Pro A : Pro T → Pro T A is monadic.

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Monadicity and definability of the binding group

Lemma (Kamensky)

ι : T0 → T is a stable immersion iff Ind ι : Ind T0 → Ind T is a cartesian closed functor.

Lemma

If A : T0 → Sets contains a basis for the internal cover ι (i.e. every Y is T0-internal over A), then ιA is an equivalence.

Lemma

If T is a complete theory, for every A the functor Pro A : Pro T → Pro T A is monadic.

Proof.

Use Duskin variant of Beck’s monadicity

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Monadicity and definability of the binding group

Lemma (Kamensky)

ι : T0 → T is a stable immersion iff Ind ι : Ind T0 → Ind T is a cartesian closed functor.

Lemma

If A : T0 → Sets contains a basis for the internal cover ι (i.e. every Y is T0-internal over A), then ιA is an equivalence.

Lemma

If T is a complete theory, for every A the functor Pro A : Pro T → Pro T A is Hopf monadic.

Proof.

See [BLV11] with a caveat.

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Galois theory of neutral internal covers

Theorem

Let ι : T0 → T be an internal cover neutralized by A. (i.e. A contains an internality basis and A0 = Γ0 = dclT0(0) ) There is a pro-group G = Pro Gk in Pro T0 and an equivalence between T and T G

0 .

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Galois theory of neutral internal covers

Theorem

Let ι : T0 → T be an internal cover neutralized by A. There is a pro-group G = Pro Gk in Pro T0 and an equivalence between T and T G

0 .

Proof.

By [BLV11], augmented Hopf monads are ⊗-representable by Hopf monoids.

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Galois theory of neutral internal covers

Theorem

Let ι : T0 → T be an internal cover neutralized by A. There is a pro-group G = Pro Gk in Pro T0 and an equivalence between T and T G

0 .

Corollary

For every M | = T A, Aut(M/M0)) ≃A lim ← − M0(Gk).

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Galois theory of neutral internal covers

Theorem

Let ι : T0 → T be an internal cover neutralized by A. There is a pro-group G = Pro Gk in Pro T0 and an equivalence between T and T G

0 .

Corollary

For every M | = T A, there are biyective correspondences between: {K : T → Sets | dclT(M0) ⊆ K ⊆ M} {H ≤ G | H = Pro Hk}

Aut(M/K)≃lim

← − M0(Hl)

MM0(Hk )

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Galois theory of neutral internal covers

Proof.

{K | dclT(M0) ⊆ K ⊆ M} {H ≤ G | H = Pro Hk} {B | dclT(0) ⊆ B ⊆ A} {T

′

− − →T ′

π

− − →T A | π′ = A}

K∩A T H dcl(M0∪B) T B Γ′′

where the ′ are stable embeddings, therefore ′ι : T0 → T ′ is an internal cover neutralized by A.

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References

Alain Brugui` eres, Steve Lack, and Alexis Virelizier, Hopf monads

• n monoidal categories, Advances in Mathematics 227 (2011),
• no. 2, 745–800.

Alexander Grothendieck, Revˆ etements ´ etales et groupe fondamental (SGA 1), Lecture Notes in Mathematics, vol. 224, Springer, 1971. Moshe Kamensky, A categorical approach to internality, Models, logics, and higher-dimensional categories (Bradd Hart et al., eds.), CRM proc. & lecture notes, vol. 53, American Math. Soc., 2011, pp. 139–156. , Model theory and the tannakian formalism, Transactions

• f the American Math. Soc. 367 (2015), no. 2, 1095–1120.

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