The Grothendieck Construction for Enriched, Internal and -Categories - PowerPoint PPT Presentation

Examples For c ∈ C , let C / c be the slice category over c : x y c g ◦− Have C / • : C → Cat sending g : c → d to C / c − − − − − − → C / d . ( C / • ) ⋊ C has objects ( x → c , c ) and morphisms: x y g c d

Examples For c ∈ C , let C / c be the slice category over c : x y c g ◦− Have C / • : C → Cat sending g : c → d to C / c − − − − − − → C / d . ( C / • ) ⋊ C has objects ( x → c , c ) and morphisms: x y g c d ( C / • ) ⋊ C = Arr C and Arr C → C is the codomain functor.

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R .

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R . Mod • ⋊ Ring op has objects ( M , R ) and morphisms:

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R . Mod • ⋊ Ring op has objects ( M , R ) and morphisms: ( ) , ( M , R ) − − − − − − − − − − − → ( N , S )

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R . Mod • ⋊ Ring op has objects ( M , R ) and morphisms: f − → S ) ( , R ( M , R ) − − − − − − − − − − − → ( N , S )

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R . Mod • ⋊ Ring op has objects ( M , R ) and morphisms: f − → S ) ( M → f ∗ N , R ( M , R ) − − − − − − − − − − − → ( N , S )

Examples We have a functor Mod • : Ring op → Cat sending f : R → S to f ∗ : Mod S → Mod R . Mod • ⋊ Ring op has objects ( M , R ) and morphisms: f − → S ) ( M → f ∗ N , R ( M , R ) − − − − − − − − − − − → ( N , S ) This is the global module category Mod .

The Skew Group Ring and Smash Products Let G be a group.

The Skew Group Ring and Smash Products Let G be a group. Instead of G acting on another group, suppose it acts on a k -algebra A G × A → A .

The Skew Group Ring and Smash Products Let G be a group. Instead of G acting on another group, suppose it acts on a k -algebra A G × A → A . Can form the skew group ring A ⋊ G

The Skew Group Ring and Smash Products Let G be a group. Instead of G acting on another group, suppose it acts on a k -algebra A G × A → A . Can form the skew group ring A ⋊ G = � g ∈ G A where ( a , g ) · ( b , h ) = ( a ( g · b ) , gh ) .

The Skew Group Ring and Smash Products Let G be a group. Instead of G acting on another group, suppose it acts on a k -algebra A G × A → A . Can form the skew group ring A ⋊ G = � g ∈ G A where ( a , g ) · ( b , h ) = ( a ( g · b ) , gh ) . But we don’t have an algebra map A ⋊ G → kG . . .

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g .

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C .

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ).

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X - grading W = � x ∈ X W x

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X - grading W = � x ∈ X W x an X - coaction W → W × X

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X - grading W = � x ∈ X W x an X - coaction W → W × X

Interlude: Comonoids and Comodules kG is both an algebra and a coalgebra , with comultiplication : ∆: kG → kG ⊗ kG , g → g ⊗ g . Can define comodules for any coalgebra C , with coactions M → M ⊗ C . We can similarly define comonoids and their comodules in any monoidal category ( V , ⊗ , 1 ). Any X ∈ Set has a unique comonoid structure, and TFAE: a function f : W → X an X - grading W = � x ∈ X W x an X - coaction W → W × X In Vect k , these are not equivalent.

The Skew Group Ring and Smash Products We don’t have an algebra map from A ⋊ G = � g ∈ G A to kG .

The Skew Group Ring and Smash Products We don’t have an algebra map from A ⋊ G = � g ∈ G A to kG . But we do have a G-grading on A ⋊ G ,

The Skew Group Ring and Smash Products We don’t have an algebra map from A ⋊ G = � g ∈ G A to kG . But we do have a G-grading on A ⋊ G , or equivalently, a kG - coaction on A ⋊ G ( a , g ) �→ ( a , g ) ⊗ g .

The Skew Group Ring and Smash Products We don’t have an algebra map from A ⋊ G = � g ∈ G A to kG . But we do have a G-grading on A ⋊ G , or equivalently, a kG - coaction on A ⋊ G ( a , g ) �→ ( a , g ) ⊗ g . The coaction perspective allows us to replace kG with any bialgebra or Hopf algebra H .

The Skew Group Ring and Smash Products Theorem (Cohen-Montgomery 1984) For G a group, there is a bijective correspondence:     ⋊  G-graded algebras  G-actions     ∼ = G × A → A    A ⋊ G  fibers  

The Skew Group Ring and Smash Products Theorem (Cohen-Montgomery 1984) For G a group, there is a bijective correspondence:     ⋊  G-graded algebras  G-actions     ∼ = G × A → A    A ⋊ G  fibers   Theorem (v.d.Bergh 1984, Blattner-Montgomery 1985) For H a Hopf algebra, there is a bijective correspondence:     ⋊  H-module algebras   H-comodule algebras      ∼ = H ⊗ A → A    A ⋊ H  coinv    

Grothendieck Smash Product Construction A ⋊ H N • ⋊ C k -linear many objects Semi-direct Product N ⋊ G

? Grothendieck Smash Product Construction A ⋊ H N • ⋊ C k -linear many objects Semi-direct Product N ⋊ G

Enriched and Internal Categories A small category C has: a set of objects C 0 for all x , y ∈ C 0 , a set of arrows Hom C ( x , y )

Enriched and Internal Categories A small category C has: a set of objects C 0 for all x , y ∈ C 0 , a set of arrows Hom C ( x , y ) Can replace ( Set , × , {∗} ) with any monoidal category ( V , ⊗ , 1 ):

Enriched and Internal Categories A small category C has: a set of objects C 0 for all x , y ∈ C 0 , a set of arrows Hom C ( x , y ) Can replace ( Set , × , {∗} ) with any monoidal category ( V , ⊗ , 1 ): A V -enriched category C has: a set of objects C 0 for all x , y ∈ C 0 , arrows Hom C ( x , y ) ∈ V

Enriched and Internal Categories A small category C has: a set of objects C 0 for all x , y ∈ C 0 , a set of arrows Hom C ( x , y ) Can replace ( Set , × , {∗} ) with any monoidal category ( V , ⊗ , 1 ): A V -enriched category C has: a set of objects C 0 for all x , y ∈ C 0 , arrows Hom C ( x , y ) ∈ V A V -internal category C has: objects C 0 ∈ V arrows C 1 ∈ V

Enriched and Internal Categories A small category C has: a set of objects C 0 for all x , y ∈ C 0 , a set of arrows Hom C ( x , y ) Can replace ( Set , × , {∗} ) with any monoidal category ( V , ⊗ , 1 ): A V -enriched category C has: a set of objects C 0 for all x , y ∈ C 0 , arrows Hom C ( x , y ) ∈ V A V -internal category C has: objects C 0 ∈ V objects C 0 ∈ Comon ( V ) arrows C 1 ∈ V arrows C 1 ∈ C 0 Comod C 0

Enriched and Internal Categories for ( Vect k , ⊗ k , k )

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y )

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗ A many-object enriched category replaces ∗ with any set.

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗ A many-object enriched category replaces ∗ with any set. Any k -linear category gives rise to a Vect k -internal category with: objects kC 0 arrows ⊕ x , y Hom C ( x , y )

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗ A many-object enriched category replaces ∗ with any set. Any k -linear category gives rise to a Vect k -internal category with: objects kC 0 arrows ⊕ x , y Hom C ( x , y ) A e.g. a k -algebra A gives an internal category k

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗ A many-object enriched category replaces ∗ with any set. Any k -linear category gives rise to a Vect k -internal category with: objects kC 0 arrows ⊕ x , y Hom C ( x , y ) A e.g. a k -algebra A gives an internal category k A ‘many-object’ internal category replaces k with a k -coalgebra.

Enriched and Internal Categories for ( Vect k , ⊗ k , k ) A Vect k -enriched category is a k -linear category C with: a set of objects C 0 for all x , y ∈ C 0 , a k -vector space Hom C ( x , y ) A e.g. a k -algebra A gives a k -linear category ∗ A many-object enriched category replaces ∗ with any set. Any k -linear category gives rise to a Vect k -internal category with: objects kC 0 arrows ⊕ x , y Hom C ( x , y ) A e.g. a k -algebra A gives an internal category k A ‘many-object’ internal category replaces k with a k -coalgebra. (possibly with other properties, e.g. cocommutativty)

cocomm. comon. of objects k -linear objects k -linear Homs Smash Grothendieck Product Construction N • ⋊ C A ⋊ H k -linear many objects Semi-direct Product N ⋊ G

cocomm. comon. of objects k -linear objects Enriched version k -linear Homs Smash Grothendieck Product Construction N • ⋊ C A ⋊ H k -linear many objects Semi-direct Product N ⋊ G

Internal version cocomm. comon. of objects k -linear objects Smash Enriched Product version A ⋊ H k -linear Homs Smash Grothendieck Product Construction N • ⋊ C A ⋊ H k -linear many objects Semi-direct Product N ⋊ G

Enriched Versions Suppose V has coproducts, and ⊗ preserves them.

Enriched Versions Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)     ⋊ C-graded V -cats  Functors    ∼ = A • : C → V - Cat A • ⋊ C     fibers

Enriched Versions Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)     ⋊ C-graded V -cats  Functors    ∼ = A • : C → V - Cat A • ⋊ C     fibers Want to replace the ordinary category C with a V -category C .

Enriched Versions Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)     ⋊ C-graded V -cats  Functors    ∼ = A • : C → V - Cat A • ⋊ C     fibers Want to replace the ordinary category C with a V -category C . Theorem (W) Let C be a comonoidal V -category.

Enriched Versions Suppose V has coproducts, and ⊗ preserves them. Theorem (Cibils-Marcos 2006, Lowen 2008, Tamaki 2009)     ⋊ C-graded V -cats  Functors    ∼ = A • : C → V - Cat A • ⋊ C     fibers Want to replace the ordinary category C with a V -category C . Theorem (W) Let C be a comonoidal V -category. Then     ⋊ C -module V -cats C -comodule V -cats     ∼ = C ⊗ A → A A ⋊ C     coinv

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them.

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then     ⋊ C -module int cats C -comod int cats     ∼ = C ⊗ A → A A ⋊ C     coinv

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then     ⋊ C -module int cats C -comod int cats     ∼ = C ⊗ A → A A ⋊ C     coinv Let C = ( C 0 , C 1 ) be comonoidal internal category, and A = ( A 0 , A 1 ) be a C -module category.

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then     ⋊ C -module int cats C -comod int cats     ∼ = C ⊗ A → A A ⋊ C     coinv Let C = ( C 0 , C 1 ) be comonoidal internal category, and A = ( A 0 , A 1 ) be a C -module category. Can form A ⋊ C with objects A 0

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then     ⋊ C -module int cats C -comod int cats     ∼ = C ⊗ A → A A ⋊ C     coinv Let C = ( C 0 , C 1 ) be comonoidal internal category, and A = ( A 0 , A 1 ) be a C -module category. Can form A ⋊ C with objects A 0 and arrows A 1 ⊠ A 0 ( C 1 ⊠ C 0 A 0 ).

Internal Version Theorem (W) Suppose V has equalizers, and ⊗ preserves them. Let C be a comonoidal internal category. Then     ⋊ C -module int cats C -comod int cats     ∼ = C ⊗ A → A A ⋊ C     coinv Let C = ( C 0 , C 1 ) be comonoidal internal category, and A = ( A 0 , A 1 ) be a C -module category. Can form A ⋊ C with objects A 0 and arrows A 1 ⊠ A 0 ( C 1 ⊠ C 0 A 0 ). When C = ( k , H ) , A = ( k , A ), this is just A ⊠ k ( H ⊠ k k ) ∼ = A ⊗ H .

Internal Version A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 ⊠ A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 1 ⊠ A 0 C 1 ⊠ C 0 C 1 ⊠ C 0 A 0 A 1 A 1 C 1 A 0 ⊠ A 0 ⊠ A 0 ⊠ C 0 A 1 C 1 A 0 ⊠ A 0 ⊠ C 0

Internal Version A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 ⊠ A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 1 ⊠ A 0 C 1 ⊠ C 0 C 1 ⊠ C 0 A 0 A 1 A 1 C 1 A 0 ⊠ A 0 ⊠ A 0 ⊠ C 0 A 1 C 1 A 0 ⊠ A 0 ⊠ C 0

Internal Version A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 ⊠ A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 1 ⊠ A 0 C 1 ⊠ C 0 C 1 ⊠ C 0 A 0 A 1 A 1 C 1 A 0 ⊠ A 0 ⊠ A 0 ⊠ C 0 A 1 C 1 A 0 ⊠ A 0 ⊠ C 0

Internal Version A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 ⊠ A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 1 ⊠ A 0 C 1 ⊠ C 0 C 1 ⊠ C 0 A 0 A 1 A 1 C 1 A 0 ⊠ A 0 ⊠ A 0 ⊠ C 0 A 1 C 1 A 0 ⊠ A 0 ⊠ C 0

Internal Version A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 ⊠ A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 0 A 1 ⊠ A 0 C 1 ⊠ C 0 A 1 ⊠ A 0 C 1 ⊠ C 0 C 1 ⊠ C 0 A 0 A 1 A 1 C 1 A 0 ⊠ A 0 ⊠ A 0 ⊠ C 0 A 1 C 1 A 0 ⊠ A 0 ⊠ C 0