Cloak and dagger Chris Heunen 1 / 34
Algebra and coalgebra Increasing generality: ◮ Vector space with bilinear (co)multiplication ◮ (Co)monoid in monoidal category ◮ (Co)monad: (co)monoid in functor category ◮ (Co)algebras for a (co)monad Interaction between algebra and coalgebra? 2 / 34
Cloak and dagger ◮ Situation involving secrecy or mystery 3 / 34
Cloak and dagger ◮ Situation involving secrecy or mystery ◮ Purpose of cloak is to obscure presence or movement of dagger 3 / 34
Cloak and dagger ◮ Situation involving secrecy or mystery ◮ Purpose of cloak is to obscure presence or movement of dagger ◮ Dagger, a concealable and silent weapon: dagger categories ◮ Cloak, worn to hide identity: Frobenius law 3 / 34
Dagger 4 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op ◮ Partial invertible computing : sets and partial injections 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op ◮ Partial invertible computing : sets and partial injections ◮ Probabilistic computing : doubly stochastic maps, f † = f T 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op ◮ Partial invertible computing : sets and partial injections ◮ Probabilistic computing : doubly stochastic maps, f † = f T ◮ Quantum computing : Hilbert spaces, f † = f T 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op ◮ Partial invertible computing : sets and partial injections ◮ Probabilistic computing : doubly stochastic maps, f † = f T ◮ Quantum computing : Hilbert spaces, f † = f T ◮ Second order : dagger functors F ( f ) † = F ( f † ) 5 / 34
Dagger Method to turn algebra into coalgebra: self-duality C op ≃ C † → C with A † = A on objects, f †† = f on maps Dagger: functor C op f † f A − → B B − → A Dagger category: category equipped with dagger ◮ Invertible computing : groupoid, f † = f − 1 ◮ Possibilistic computing : sets and relations, R † = R op ◮ Partial invertible computing : sets and partial injections ◮ Probabilistic computing : doubly stochastic maps, f † = f T ◮ Quantum computing : Hilbert spaces, f † = f T ◮ Second order : dagger functors F ( f ) † = F ( f † ) ◮ Unitary representations : [ G, Hilb ] † 5 / 34
Never bring a knife to a gun fight ◮ Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) 6 / 34
Never bring a knife to a gun fight ◮ Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) ◮ Evil: demand equality A † = A of objects “Homotopy type theory” Univalent foundations program, 2013 6 / 34
Never bring a knife to a gun fight ◮ Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) ◮ Evil: demand equality A † = A of objects ◮ Dagger category theory different beast: isomorphism is not the correct notion of ‘sameness’ “Homotopy type theory” Univalent foundations program, 2013 6 / 34
Way of the dagger Motto: “ everything in sight ought to cooperate with the dagger ” 7 / 34
Way of the dagger Motto: “ everything in sight ought to cooperate with the dagger ” ◮ dagger isomorphism (unitary): f − 1 = f † ◮ dagger monic (isometry): f † ◦ f = id 7 / 34
Way of the dagger Motto: “ everything in sight ought to cooperate with the dagger ” ◮ dagger isomorphism (unitary): f − 1 = f † ◮ dagger monic (isometry): f † ◦ f = id ◮ dagger equalizer / kernel / biproduct 7 / 34
Way of the dagger Motto: “ everything in sight ought to cooperate with the dagger ” ◮ dagger isomorphism (unitary): f − 1 = f † ◮ dagger monic (isometry): f † ◦ f = id ◮ dagger equalizer / kernel / biproduct ◮ monoidal dagger category: ( f ⊗ g ) † = f † ⊗ g † , α − 1 = α † 7 / 34
Way of the dagger Motto: “ everything in sight ought to cooperate with the dagger ” ◮ dagger isomorphism (unitary): f − 1 = f † ◮ dagger monic (isometry): f † ◦ f = id ◮ dagger equalizer / kernel / biproduct ◮ monoidal dagger category: ( f ⊗ g ) † = f † ⊗ g † , α − 1 = α † What about monoids?? 7 / 34
Cloaks are worn 8 / 34
Frobenius algebra Many definitions over a field k : ◮ algebra A with functional A → k , kernel without left ideals 9 / 34
Frobenius algebra Many definitions over a field k : ◮ algebra A with functional A → k , kernel without left ideals ◮ algebra A with finitely many minimal right ideals 9 / 34
Frobenius algebra Many definitions over a field k : ◮ algebra A with functional A → k , kernel without left ideals ◮ algebra A with finitely many minimal right ideals ◮ algebra A with nondegenerate A ⊗ A → k with [ ab, c ] = [ a, bc ] 9 / 34
Frobenius algebra Many definitions over a field k : ◮ algebra A with functional A → k , kernel without left ideals ◮ algebra A with finitely many minimal right ideals ◮ algebra A with nondegenerate A ⊗ A → k with [ ab, c ] = [ a, bc ] ◮ algebra A with comultiplication δ : A → A ⊗ A satisfying (id ⊗ µ ) ◦ ( δ ⊗ id) = ( µ ⊗ id) ◦ (id ⊗ δ ) 9 / 34
Frobenius algebra Many definitions over a field k : ◮ algebra A with functional A → k , kernel without left ideals ◮ algebra A with finitely many minimal right ideals ◮ algebra A with nondegenerate A ⊗ A → k with [ ab, c ] = [ a, bc ] ◮ algebra A with comultiplication δ : A → A ⊗ A satisfying (id ⊗ µ ) ◦ ( δ ⊗ id) = ( µ ⊗ id) ◦ (id ⊗ δ ) ◮ algebra A with equivalent left and right regular representations “Theorie der hyperkomplexen Gr¨ oßen I” Sitzungsberichte der Preussischen Akademie der Wissenschaften 504–537, 1903 “On Frobeniusean algebras II” Annals of Mathematics 42(1):1–21, 1941 9 / 34
Frobenius law in algebra Any finite group G induces Frobenius group algebra A : ◮ A has orthonormal basis { g ∈ G } ◮ multiplication g ⊗ h �→ gh ◮ unit e 10 / 34
Frobenius law in algebra Any finite group G induces Frobenius group algebra A : ◮ A has orthonormal basis { g ∈ G } ◮ multiplication g ⊗ h �→ gh ◮ unit e h gh − 1 ⊗ h ◮ comultiplication g �→ � k gk − 1 ⊗ kh on g ⊗ h ◮ both sides of Frobenius law evaluate to � So Frobenius algebra incorporates finite group representation theory 10 / 34
Frobenius law in algebra Frobenius algebras are wonderful: ◮ left and right Artinian ◮ left and right self-injective 11 / 34
Frobenius law in algebra Frobenius algebras are wonderful: ◮ left and right Artinian ◮ left and right self-injective ◮ Frobenius property is independent of base field k ! ◮ Extension of scalars: if l extends k , then A Frobenius over k iff l ⊗ k A Frobenius over l ◮ Restriction of scalars: if l extends k , then A Frobenius over l iff A Frobenius over k 11 / 34
Frobenius law in mathematics ◮ Number theory: commutative Frobenius algebras are Gorenstein “Modular elliptic curves and Fermat’s last theorem” Annals of Mathematics 142(3):443–551, 1995 12 / 34
Recommend
More recommend
