Dagger Category Theory Chris Heunen and Martti Karvonen 1 / 19 Outline I What are dagger categories? I What are dagger monads? I What are dagger limits? I What are evils about daggers? 2 / 19 y Terminology: adjoints in Hilbert spaces hf(x) j yiY = hx j f (y)iX If S(X) is poset of closed subspaces, get S(f): S(X)op ! S(Y ) Theorem [Palmquist 74]: S(f) and S(f y) adjoint, and up to scalar any adjunction of this form Dagger A dagger is contravariant involutive identity-on-objects endofunctor f = f yy X Y f y 3 / 19 Dagger A dagger is contravariant involutive identity-on-objects endofunctor f = f yy X Y f y y Terminology: adjoints in Hilbert spaces hf(x) j yiY = hx j f (y)iX If S(X) is poset of closed subspaces, get S(f): S(X)op ! S(Y ) Theorem [Palmquist 74]: S(f) and S(f y) adjoint, and up to scalar any adjunction of this form 3 / 19 I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid 4 / 19 I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices 4 / 19 I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions 4 / 19 I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections 4 / 19 I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ 4 / 19 I Dagger functors and natural transformations I Unitary representations and intertwiners Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y 4 / 19 Examples I Any groupoid I Hilbert spaces and continuous linear maps I Sets and relations I Finite sets and doubly stochastic matrices I Dagger categories and contravariant adjunctions I Inverse category: any f has unique g with f = gfg and g = fgf I Sets and partial injections I Free dagger category: same objects, [X · ! · · · · ! Y ]∼ I Cofree dagger category: same objects, pairs X Y I Dagger functors and natural transformations I Unitary representations and intertwiners 4 / 19 monad ? limit ? isn't this trivially trivial? Way of the dagger Category theory Dagger category theory isomorphism unitary f −1 = f y idempotent projection f = f y ◦ f functor dagger functor F (f y) = F (f)y y y natural transform natural transformation (α )X = (αX ) monoidal structure monoidal dagger structure (f ⊗ g)y = f y ⊗ gy 5 / 19 isn't this trivially trivial? Way of the dagger Category theory Dagger category theory isomorphism unitary f −1 = f y idempotent projection f = f y ◦ f functor dagger functor F (f y) = F (f)y y y natural transform natural transformation (α )X = (αX ) monoidal structure monoidal dagger structure (f ⊗ g)y = f y ⊗ gy monad ? limit ? 5 / 19 Way of the dagger Category theory Dagger category theory isomorphism unitary f −1 = f y idempotent projection f = f y ◦ f functor dagger functor F (f y) = F (f)y y y natural transform natural transformation (α )X = (αX ) monoidal structure monoidal dagger structure (f ⊗ g)y = f y ⊗ gy monad ? limit ? isn't this trivially trivial? 5 / 19 I 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 I Principle: if P =) Q for categories, then P y + laws =) Qy + laws for dagger categories Formal dagger category theory I Daggers not preserved under equivalence 6 / 19 I Principle: if P =) Q for categories, then P y + laws =) Qy + laws for dagger categories Formal dagger category theory I Daggers not preserved under equivalence I 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 I Daggers not preserved under equivalence I 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 I Principle: if P =) Q for categories, then P y + laws =) Qy + laws for dagger categories 6 / 19 Kl(GF ) D FEM(GF ) F G C I Dagger adjunction is adjunction in DagCat: no left/right I Dagger monad should at least be dagger functor: so comonad I What interaction between monad and comonad? Dagger monads dagger monads monads I Want = dagger adjunctions adjunctions 7 / 19 I Dagger monad should at least be dagger functor: so comonad I What interaction between monad and comonad? Dagger monads dagger monads monads I Want = dagger adjunctions adjunctions Kl(GF ) D FEM(GF ) F G C I Dagger adjunction is adjunction in DagCat: no left/right 7 / 19 Dagger monads dagger monads monads I Want = dagger adjunctions adjunctions Kl(GF ) D FEM(GF ) F G C I Dagger adjunction is adjunction in DagCat: no left/right I Dagger monad should at least be dagger functor: so comonad I What interaction between monad and comonad? 7 / 19 I If M is dagger Frobenius monoid, then − ⊗ M is dagger monad I Dagger adjunctions induce dagger monads Dagger monads I A dagger monad is a monad that is a dagger functor satisfying µT ◦ T µy = T µ ◦ µyT = 8 / 19 I Dagger adjunctions induce dagger monads Dagger monads I A dagger monad is a monad that is a dagger functor satisfying µT ◦ T µy = T µ ◦ µyT = I If M is dagger Frobenius monoid, then − ⊗ M is dagger monad 8 / 19 Dagger monads I A dagger monad is a monad that is a dagger functor satisfying µT ◦ T µy = T µ ◦ µyT = I If M is dagger Frobenius monoid, then − ⊗ M is dagger monad I Dagger adjunctions induce dagger monads 8 / 19 I Frobenius law for monoid M is Frobenius law for monad − ⊗ M = Kleisli algebras I If T is dagger monad on C, then Kl(T ) has dagger y T (f y) A f T (B) 7! B η T (B) µ T 2(B) T (A) that commutes with C ! Kl(T ) and Kl(T ) ! C 9 / 19 Kleisli algebras I If T is dagger monad on C, then Kl(T ) has dagger y T (f y) A f T (B) 7! B η T (B) µ T 2(B) T (A) that commutes with C ! Kl(T ) and Kl(T ) ! C I Frobenius law for monoid M is Frobenius law for monad − ⊗ M = 9 / 19 I Largest full subcategory with Kl(T ) and EM(T ) ! C dagger I There are EM-algebras that are not FEM Eilenberg-Moore algebras a I Frobenius-Eilenberg-Moore algebra is algebra T (A) ! A with T (a)y T (A) T 2(A) µy µ T 2(A) T (A) T (a) Gives full subcategory FEM(T ) 10 / 19 I There are EM-algebras that are not FEM Eilenberg-Moore algebras a I Frobenius-Eilenberg-Moore algebra is algebra T (A) ! A with T (a)y T (A) T 2(A) µy µ T 2(A) T (A) T (a) Gives full subcategory FEM(T ) I Largest full subcategory with Kl(T ) and EM(T ) ! C dagger 10 / 19 Eilenberg-Moore algebras a I Frobenius-Eilenberg-Moore algebra is algebra T (A) ! A with T (a)y T (A) T 2(A) µy µ T 2(A) T (A) T (a) Gives full subcategory FEM(T ) I Largest full subcategory with Kl(T ) and EM(T ) ! C dagger I There are EM-algebras that are not FEM 10 / 19 Proof.