Galois theory, a logical path from Grothendiecks version to the - PowerPoint PPT Presentation

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 over 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 over 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 over constructible sets. 2 of 8

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 . Aut( E / K ) { K | F ≤ K ≤ E } { H ≤ G | H is closed } E H Grothendieck’s version Let ω : C → S ets f be a fundamental functor from a Galoisian category, then π = Aut( ω ) is a pro-finite group and ω lifts to an 3 of 8 equivalence between C and the category of finite continuous π -actions S ets π .

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 → S ets f 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 S ets π f . S ets π C f ω S ets f 3 of 8

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 → S ets f 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 S ets π f . In particular , let C op be the category of finite ´ etale F -algebras split by E and take ω ( X ) = X ( E ) = Hom F –alg. ( A , E ) when X = Spec A in C , then Aut( ω ) = Aut( E / F ) and C is equivalent to S ets G f . 3 of 8

b b Categorical logic Given a first order theory T V 0 � The category of T -definables V (including imaginary sorts) µ × 0 T = Def( T eq ) is a boolean K pre-topos . � Models are logical functors Function µ : K × V → V . Constant 0 ∈ V M : T → S ets. Q � Interpretations are logical functors ι : T 0 → T . < × Q � ι is an immersion if ι X : Sub T 0 ( X ) → Sub T ( ι X ) is A binary relation < in the sort Q . an isomorphism for every X . 4 of 8

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Categorical logic Given a first order theory T V 0 � The category of T -definables V (including imaginary sorts) µ × 0 T = Def( T eq ) is a boolean K pre-topos . � Models are logical functors Function µ : K × V → V . Constant 0 ∈ V M : T → S ets. Q � Interpretations are logical functors ι : T 0 → T . < × Q � ι is an immersion if ι X : Sub T 0 ( X ) → Sub T ( ι X ) is A binary relation < in the sort Q . an isomorphism for every X . 4 of 8

Categorical logic Given a first order theory T Def( T ) � The category of T -definables V (including imaginary sorts) V T = Def( T eq ) is a boolean µ × pre-topos . K � Models are logical functors Def( T 0 ) M : T → S ets. � Interpretations are logical Every M | = T induces a functors ι : T 0 → T . M 0 | = T 0 . � ι is an immersion if ι X : Sub T 0 ( X ) → Sub T ( ι X ) is an isomorphism for every X . 4 of 8

Categorical logic Given a first order theory T Def( T ) � The category of T -definables V (including imaginary sorts) V T = Def( T eq ) is a boolean µ × pre-topos . K � Models are logical functors Def( T 0 ) M : T → S ets. � Interpretations are logical Every M | = T induces a functors ι : T 0 → T . M 0 | = T 0 . � ι is an immersion if ι X : Sub T 0 ( X ) → Sub T ( ι X ) is an isomorphism for every X . 4 of 8

Categorical logic Given a first order theory T Def( T ) � The category of T -definables V (including imaginary sorts) V T = Def( T eq ) is a boolean µ × pre-topos . K � Models are logical functors Def( T 0 ) M : T → S ets. � Interpretations are logical Every M | = T induces a functors ι : T 0 → T . M 0 | = T 0 . In fact, ι ∗ : Mod( T ) → Mod( T 0 ) is an � ι is an immersion if equivalence, if and only if, ι is an ι X : Sub T 0 ( X ) → Sub T ( ι X ) is equivalence. an isomorphism for every X . 4 of 8

Definable closure and internal covers � Given A ⊆ M | = T , the functor A = dcl( A ) preserves limits and  A co-limits, but not necessarily T A = Def( T , A ) T images (i.e. ∃ quantifier). Γ A =Hom( 1 , ?) � ι : T 0 → T is a stable A =dcl( A ) immersion if ι A is an immersion S ets for every A . � Y is T 0 -internal over A if for Definition = T A , ι is an internal cover if it’s stable every M | and every T -definable is M ( Y ) = dcl( M 0 ∪ A ) T 0 -internal. 5 of 8

Definable closure and internal covers � Given A ⊆ M | = T , the functor A = dcl( A ) preserves limits and  A co-limits, but not necessarily T A = Def( T , A ) T images (i.e. ∃ quantifier). Γ A =Hom( 1 , ?) � ι : T 0 → T is a stable A =dcl( A ) immersion if ι A is an immersion S ets for every A . � Y is T 0 -internal over A if for  A 0 T A 0 = T A , T 0 every M | 0 ι ι A M ( Y ) = dcl( M 0 ∪ A )  A T A T 5 of 8 Definition

b Definable closure and internal covers � Given A ⊆ M | = T , the functor Def( T ) A = dcl( A ) preserves limits and co-limits, but not necessarily B b Z b images (i.e. ∃ quantifier). � ι : T 0 → T is a stable immersion if ι A is an immersion for every A . Def( T 0 ) X � Y is T 0 -internal over A if for = T A , every M | Definition M ( Y ) = dcl( M 0 ∪ A ) ι is an internal cover if it’s stable and every T -definable is T 0 -internal. 5 of 8

b Definable closure and internal covers � Given A ⊆ M | = T , the functor Def( T ) A = dcl( A ) preserves limits and co-limits, but not necessarily B b Z b images (i.e. ∃ quantifier). � ι : T 0 → T is a stable immersion if ι A is an immersion for every A . A a Def( T 0 ) X b a � Y is T 0 -internal over A if for Z ′ = T A , every M | Definition M ( Y ) = dcl( M 0 ∪ A ) ι is an internal cover if it’s stable and every T -definable is T 0 -internal. 5 of 8