Dagger Category Theory Chris Heunen and Martti Karvonen 1 / 19 - PowerPoint PPT Presentation

Dagger Category Theory Chris Heunen and Martti Karvonen 1 / 19

Outline ◮ What are dagger categories? ◮ What are dagger monads? ◮ What are dagger limits? ◮ What are evils about daggers? 2 / 19

Dagger A dagger is contravariant involutive identity-on-objects endofunctor f = f †† X Y f † 3 / 19

Dagger A dagger is contravariant involutive identity-on-objects endofunctor f = f †† X Y f † Terminology: adjoints in Hilbert spaces � f ( x ) | y � Y = � x | f † ( y ) � X If S ( X ) is poset of closed subspaces, get S ( f ): S ( X ) op → S ( Y ) Theorem [Palmquist 74]: S ( f ) and S ( f † ) adjoint, and up to scalar any adjunction of this form 3 / 19

Examples ◮ Any groupoid 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices ◮ Dagger categories and contravariant adjunctions 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices ◮ Dagger categories and contravariant adjunctions ◮ Inverse category: any f has unique g with f = gfg and g = fgf ◮ Sets and partial injections 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices ◮ Dagger categories and contravariant adjunctions ◮ Inverse category: any f has unique g with f = gfg and g = fgf ◮ Sets and partial injections ◮ Free dagger category: same objects, [ X ← · → · · · ← · → Y ] ∼ 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices ◮ Dagger categories and contravariant adjunctions ◮ Inverse category: any f has unique g with f = gfg and g = fgf ◮ Sets and partial injections ◮ Free dagger category: same objects, [ X ← · → · · · ← · → Y ] ∼ ◮ Cofree dagger category: same objects, pairs X ⇆ Y 4 / 19

Examples ◮ Any groupoid ◮ Hilbert spaces and continuous linear maps ◮ Sets and relations ◮ Finite sets and doubly stochastic matrices ◮ Dagger categories and contravariant adjunctions ◮ Inverse category: any f has unique g with f = gfg and g = fgf ◮ Sets and partial injections ◮ Free dagger category: same objects, [ X ← · → · · · ← · → Y ] ∼ ◮ Cofree dagger category: same objects, pairs X ⇆ Y ◮ Dagger functors and natural transformations ◮ Unitary representations and intertwiners 4 / 19

Way of the dagger Category theory Dagger category theory unitary f − 1 = f † isomorphism projection f = f † ◦ f idempotent dagger functor F ( f † ) = F ( f ) † functor natural transformation ( α † ) X = ( α X ) † natural transform monoidal dagger structure ( f ⊗ g ) † = f † ⊗ g † monoidal structure 5 / 19

Way of the dagger Category theory Dagger category theory unitary f − 1 = f † isomorphism projection f = f † ◦ f idempotent dagger functor F ( f † ) = F ( f ) † functor natural transformation ( α † ) X = ( α X ) † natural transform monoidal dagger structure ( f ⊗ g ) † = f † ⊗ g † monoidal structure monad ? limit ? 5 / 19

Way of the dagger Category theory Dagger category theory unitary f − 1 = f † isomorphism projection f = f † ◦ f idempotent dagger functor F ( f † ) = F ( f ) † functor natural transformation ( α † ) X = ( α X ) † natural transform monoidal dagger structure ( f ⊗ g ) † = f † ⊗ g † monoidal structure monad ? limit ? isn’t this trivially trivial? 5 / 19

Formal dagger category theory ◮ Daggers not preserved under equivalence 6 / 19

Formal dagger category theory ◮ Daggers not preserved under equivalence ◮ Dagger categories, dagger functors, and natural transformations: not just 2-category, but dagger 2-category 2-cells have dagger, so should have unitary coherence laws 6 / 19

Formal dagger category theory ◮ Daggers not preserved under equivalence ◮ Dagger categories, dagger functors, and natural transformations: not just 2-category, but dagger 2-category 2-cells have dagger, so should have unitary coherence laws ◮ Principle: if P = ⇒ Q for categories, then P † + laws = ⇒ Q † + laws for dagger categories 6 / 19

Dagger monads dagger monads monads ◮ Want dagger adjunctions = adjunctions 7 / 19

Dagger monads dagger monads monads ◮ Want dagger adjunctions = adjunctions Kl( GF ) FEM( GF ) D F G C ◮ Dagger adjunction is adjunction in DagCat : no left/right 7 / 19

Dagger monads dagger monads monads ◮ Want dagger adjunctions = adjunctions Kl( GF ) FEM( GF ) D F G C ◮ Dagger adjunction is adjunction in DagCat : no left/right ◮ Dagger monad should at least be dagger functor: so comonad ◮ What interaction between monad and comonad? 7 / 19

Dagger monads ◮ A dagger monad is a monad that is a dagger functor satisfying µT ◦ Tµ † = Tµ ◦ µ † T = 8 / 19

Dagger monads ◮ A dagger monad is a monad that is a dagger functor satisfying µT ◦ Tµ † = Tµ ◦ µ † T = ◮ If M is dagger Frobenius monoid, then − ⊗ M is dagger monad 8 / 19

Dagger monads ◮ A dagger monad is a monad that is a dagger functor satisfying µT ◦ Tµ † = Tµ ◦ µ † T = ◮ If M is dagger Frobenius monoid, then − ⊗ M is dagger monad ◮ Dagger adjunctions induce dagger monads 8 / 19

Kleisli algebras ◮ If T is dagger monad on C , then Kl( T ) has dagger A f T ( B ) B η T ( B ) µ † T ( f † ) T ( A ) T 2 ( B ) � � � � �→ that commutes with C → Kl( T ) and Kl( T ) → C 9 / 19

Kleisli algebras ◮ If T is dagger monad on C , then Kl( T ) has dagger A f T ( B ) B η T ( B ) µ † T ( f † ) T ( A ) T 2 ( B ) � � � � �→ that commutes with C → Kl( T ) and Kl( T ) → C ◮ Frobenius law for monoid M is Frobenius law for monad − ⊗ M = 9 / 19

Eilenberg-Moore algebras ◮ Frobenius-Eilenberg-Moore algebra is algebra T ( A ) a → A with T ( a ) † T 2 ( A ) T ( A ) µ † µ T 2 ( A ) T ( A ) T ( a ) Gives full subcategory FEM( T ) 10 / 19

Eilenberg-Moore algebras ◮ Frobenius-Eilenberg-Moore algebra is algebra T ( A ) a → A with T ( a ) † T 2 ( A ) T ( A ) µ † µ T 2 ( A ) T ( A ) T ( a ) Gives full subcategory FEM( T ) ◮ Largest full subcategory with Kl( T ) and EM( T ) → C dagger 10 / 19

Eilenberg-Moore algebras ◮ Frobenius-Eilenberg-Moore algebra is algebra T ( A ) a → A with T ( a ) † T 2 ( A ) T ( A ) µ † µ T 2 ( A ) T ( A ) T ( a ) Gives full subcategory FEM( T ) ◮ Largest full subcategory with Kl( T ) and EM( T ) → C dagger ◮ There are EM-algebras that are not FEM 10 / 19

Dagger monads Theorem If F, G are dagger adjoint, there are unique dagger functors with K J Kl( GF ) FEM( GF ) D F G C J is full, K is full and faithful, and JK is the canonical inclusion 11 / 19

Dagger monads Theorem If F, G are dagger adjoint, there are unique dagger functors with K J Kl( GF ) FEM( GF ) D F G C J is full, K is full and faithful, and JK is the canonical inclusion Proof. ◮ EM-algebra ( A, a ) is FEM iff a † is morphism ( A, a ) → ( TA, µ A ) 11 / 19

Dagger monads Theorem If F, G are dagger adjoint, there are unique dagger functors with K J Kl( GF ) FEM( GF ) D F G C J is full, K is full and faithful, and JK is the canonical inclusion Proof. ◮ EM-algebra ( A, a ) is FEM iff a † is morphism ( A, a ) → ( TA, µ A ) TA, µ A ) a ◮ ( A, a ) ∈ Im( J ) associative = ⇒ � → ( A, a ) � ∈ Im( J ) ⇒ a † ∈ Im( J ) = = ⇒ ( A, a ) ∈ FEM( GF ) 11 / 19

Strength ◮ Monad T is strong when coherent natural A ⊗ T ( B ) → T ( A ⊗ B ) ◮ monoids in C ≃ monads on C M �→ − ⊗ M T ( I ) ← � T 12 / 19

Strength ◮ Monad T is strong when coherent natural A ⊗ T ( B ) → T ( A ⊗ B ) ◮ monoids in C ≃ monads on C M �→ − ⊗ M T ( I ) ← � T ◮ Dagger monad is strong when strength is unitary ◮ Frobenius monoids in C ≃ strong dagger monads on C M �→ − ⊗ M ← � T ( I ) T 12 / 19

Strength ◮ Monad T is strong when coherent natural A ⊗ T ( B ) → T ( A ⊗ B ) ◮ monoids in C ≃ monads on C M �→ − ⊗ M T ( I ) ← � T ◮ Dagger monad is strong when strength is unitary ◮ Frobenius monoids in C ≃ strong dagger monads on C M �→ − ⊗ M ← � T ( I ) T ◮ [ Z , FHilb ] → [ N , FHilb ] has dagger adjoint f �→ Im( f ) but induced monad decreases dimension so not strong 12 / 19

Strength ◮ Monad T is strong when coherent natural A ⊗ T ( B ) → T ( A ⊗ B ) ◮ monoids in C ≃ monads on C M �→ − ⊗ M T ( I ) ← � T ◮ Dagger monad is strong when strength is unitary ◮ Frobenius monoids in C ≃ strong dagger monads on C M �→ − ⊗ M ← � T ( I ) T ◮ [ Z , FHilb ] → [ N , FHilb ] has dagger adjoint f �→ Im( f ) but induced monad decreases dimension so not strong ◮ If T commutative, then Kl( T ) dagger symmetric monoidal 12 / 19