The Existential Completion Davide Trotta University of Trento 9-7-2019
Introduction ◮ Let C be a category with finite products. A primary doctrine is � InfSL from the opposite of the a functor P : C op category C to the category of inf-semilattices;
Introduction ◮ Let C be a category with finite products. A primary doctrine is � InfSL from the opposite of the a functor P : C op category C to the category of inf-semilattices; � InfSL is elementary if for ◮ a primary doctrine P : C op every A and C in C , the functor � P ( C × A ) P id C × ∆ A : P ( C × A × A ) has a left adjoint E id C × ∆ A and these satisfy Frobenius reciprocity;
Introduction ◮ Let C be a category with finite products. A primary doctrine is � InfSL from the opposite of the a functor P : C op category C to the category of inf-semilattices; � InfSL is elementary if for ◮ a primary doctrine P : C op every A and C in C , the functor � P ( C × A ) P id C × ∆ A : P ( C × A × A ) has a left adjoint E id C × ∆ A and these satisfy Frobenius reciprocity; � InfSL is existential if, for ◮ a primary doctrine P : C op � A i , every A 1 , A 2 in C , for any projection pr i : A 1 × A 2 i = 1 , 2, the functor � P ( A 1 × A 2 ) P pr i : P ( A i ) has a left adjoint E pr i , and these satisfy Beck-Chevalley condition and Frobenius reciprocity.
Introduction The category of primary doctrines PD is a 2-category, where:
� � � � Introduction The category of primary doctrines PD is a 2-category, where: ◮ a 1-cell is a pair ( F , b ) C op P F op InfSL b R D op � D is a functor preserving products, and such that F : C � R ◦ F op is a natural transformation. b : P
� � � � Introduction The category of primary doctrines PD is a 2-category, where: ◮ a 1-cell is a pair ( F , b ) C op P F op InfSL b R D op � D is a functor preserving products, and such that F : C � R ◦ F op is a natural transformation. b : P � G such that ◮ a 2-cell is a natural transformation θ : F for every A in C and every α in PA , we have b A ( α ) ≤ R θ A ( c A ( α )) .
Examples Subobjects. C has finite limits. � InfSL . Sub C : C op The functor assigns to an object A in C the poset Sub C ( A ) of f � A the morphism subobjects of A in C and, for an arrow B � Sub C ( B ) is given by pulling a subobject Sub C ( f ): Sub C ( A ) back along f .
Examples Subobjects. C has finite limits. � InfSL . Sub C : C op The functor assigns to an object A in C the poset Sub C ( A ) of f � A the morphism subobjects of A in C and, for an arrow B � Sub C ( B ) is given by pulling a subobject Sub C ( f ): Sub C ( A ) back along f . Weak Subobjects. D has finite products and weak pullbacks. � InfSL . Ψ D : D op Ψ D ( A ) is the poset reflection of the slice category D / A , and for an f � A , the morphism Ψ D ( f ): Ψ D ( A ) � Ψ D ( B ) is arrow B g � A with f . given by a weak pullback of an arrow X
The Existential Completion � InfSL be a primary doctrine and let A ⊂ C 1 be Let P : C op the class of projections. For every object A of C consider we define P e ( A ) the following poset:
The Existential Completion � InfSL be a primary doctrine and let A ⊂ C 1 be Let P : C op the class of projections. For every object A of C consider we define P e ( A ) the following poset: g ∈A � A , α ∈ PB ); ◮ the objects are pairs ( B
� � The Existential Completion � InfSL be a primary doctrine and let A ⊂ C 1 be Let P : C op the class of projections. For every object A of C consider we define P e ( A ) the following poset: g ∈A � A , α ∈ PB ); ◮ the objects are pairs ( B h ∈A � A , α ∈ PB ) ≤ ( D f ∈A � A , γ ∈ PD ) if there ◮ ( B � D such that exists w : B B w h � A D f commutes and α ≤ P w ( γ ).
� � The Existential Completion � B in C , we define Given a morphism f : A g ∈A � B , β ∈ PC ) := ( D g ∗ f ∈A � A , P f ∗ g ( β ) ∈ PD ) P e f ( C where g ∗ f � A D f ∗ g f � B C g is a pullback.
The Existential Completion Theorem � B of A , let Given a morphism f : A h ∈A � A , α ∈ PC ) := ( C fh ∈A � B , α ∈ PC ) e E f ( C h ∈A � A , α ∈ PC ) is in P e ( A ) . Then e when ( C E f is left adjoint to P e f .
The Existential Completion Theorem � B of A , let Given a morphism f : A h ∈A � A , α ∈ PC ) := ( C fh ∈A � B , α ∈ PC ) e E f ( C h ∈A � A , α ∈ PC ) is in P e ( A ) . Then e when ( C E f is left adjoint to P e f . Theorem � InfSL be a primary doctrine, then the doctrine Let P : C op P e : C op � InfSL is existential.
The Existential Completion Theorem Consider the category PD ( P , R ) . We define � ED ( P e , R e ) E P , R : PD ( P , R ) as follow: ◮ for every 1-cell ( F , b ) , E P , R ( F , b ) := ( F , b e ) , where g � R e FA sends an object ( C � A , α ) in the b e A : P e A Fg � FA , b C ( α )) ; object ( FC � ( G , c ) , E P , R θ is essentially the ◮ for every 2-cell θ : ( F , b ) same. With the previous assignment E is a 2-functor and it is 2-adjoint to the forgetful functor.
The Existential Completion Theorem � PD is lax-idempotent; ◮ The 2-monad T e : PD
The Existential Completion Theorem � PD is lax-idempotent; ◮ The 2-monad T e : PD ◮ T e - Alg ≡ ED .
Exact Completion Theorem � InfSL , the doctrine For every elementary doctrine P : C op P e : C op � InfSL is elementary and existential.
Exact Completion Theorem � InfSL , the doctrine For every elementary doctrine P : C op P e : C op � InfSL is elementary and existential. Theorem The 2-functor Xct → PED that takes an exact category to the elementary doctrine of its subobjects has a left biadjoint which associates the exact category T P e to an elementary doctrine � InfSL . P : C op
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