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Cloak and Dagger Cloak and dagger Chris Heunen 1 / 34 Algebra and coalgebra Increasing generality: I Vector space with bilinear (co)multiplication I (Co)monoid in monoidal category I (Co)monad: (co)monoid in functor category I (Co)algebras for a (co)monad Interaction between algebra and coalgebra? 2 / 34 I Purpose of cloak is to obscure presence or movement of dagger I Dagger, a concealable and silent weapon: dagger categories I Cloak, worn to hide identity: Frobenius law Cloak and dagger I Situation involving secrecy or mystery 3 / 34 I Dagger, a concealable and silent weapon: dagger categories I Cloak, worn to hide identity: Frobenius law Cloak and dagger I Situation involving secrecy or mystery I Purpose of cloak is to obscure presence or movement of dagger 3 / 34 Cloak and dagger I Situation involving secrecy or mystery I Purpose of cloak is to obscure presence or movement of dagger I Dagger, a concealable and silent weapon: dagger categories I Cloak, worn to hide identity: Frobenius law 3 / 34 Dagger 4 / 34 y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C 5 / 34 y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger 5 / 34 y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f 5 / 34 I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R 5 / 34 y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections 5 / 34 y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f 5 / 34 y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f 5 / 34 I Unitary representations:[G; Hilb]y Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) 5 / 34 Dagger Method to turn algebra into coalgebra: self-duality Cop ' C y Dagger: functor Cop ! C with Ay = A on objects, f yy = f on maps f f y A −! BB −! A Dagger category: category equipped with dagger y −1 I Invertible computing: groupoid, f = f y op I Possibilistic computing: sets and relations, R = R I Partial invertible computing: sets and partial injections y T I Probabilistic computing: doubly stochastic maps, f = f y T I Quantum computing: Hilbert spaces, f = f y y I Second order: dagger functors F (f) = F (f ) I Unitary representations:[G; Hilb]y 5 / 34 y I Evil: demand equality A = A of objects I Dagger category theory different beast: isomorphism is not the correct notion of `sameness' \Homotopy type theory" Univalent foundations program, 2013 Never bring a knife to a gun fight I Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) 6 / 34 I Dagger category theory different beast: isomorphism is not the correct notion of `sameness' Never bring a knife to a gun fight I Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) y I Evil: demand equality A = A of objects \Homotopy type theory" Univalent foundations program, 2013 6 / 34 Never bring a knife to a gun fight I Terminology after (physics) notation (but beats identity-on-objects-involutive-contravariant-functor) y I Evil: demand equality A = A of objects I Dagger category theory different beast: isomorphism is not the correct notion of `sameness' \Homotopy type theory" Univalent foundations program, 2013 6 / 34 −1 y I dagger isomorphism (unitary): f = f y I dagger monic (isometry): f ◦ f = id I dagger equalizer / kernel / biproduct y y y −1 y I monoidal dagger category: (f ⊗ g) = f ⊗ g , α = α What about monoids?? Way of the dagger Motto: \everything in sight ought to cooperate with the dagger" 7 / 34 I dagger equalizer / kernel / biproduct y y y −1 y I monoidal dagger category: (f ⊗ g) = f ⊗ g , α = α What about monoids?? Way of the dagger Motto: \everything in sight ought to cooperate with the dagger" −1 y I dagger isomorphism (unitary): f = f y I dagger monic (isometry): f ◦ f = id 7 / 34 y y y −1 y I monoidal dagger category: (f ⊗ g) = f ⊗ g , α = α What about monoids?? Way of the dagger Motto: \everything in sight ought to cooperate with the dagger" −1 y I dagger isomorphism (unitary): f = f y I dagger monic (isometry): f ◦ f = id I dagger equalizer / kernel / biproduct 7 / 34 What about monoids?? Way of the dagger Motto: \everything in sight ought to cooperate with the dagger" −1 y I dagger isomorphism (unitary): f = f y I dagger monic (isometry): f ◦ f = id I dagger equalizer / kernel / biproduct y y y −1 y I monoidal dagger category: (f ⊗ g) = f ⊗ g , α = α 7 / 34 Way of the dagger Motto: \everything in sight ought to cooperate with the dagger" −1 y I dagger isomorphism (unitary): f = f y I dagger monic (isometry): f ◦ f = id I dagger equalizer / kernel / biproduct y y y −1 y I monoidal dagger category: (f ⊗ g) = f ⊗ g , α = α What about monoids?? 7 / 34 Cloaks are worn 8 / 34 I algebra A with finitely many minimal right ideals I algebra A with nondegenerate A ⊗ A ! k with [ab; c] = [a; bc] I algebra A with comultiplication δ : A ! A ⊗ A satisfying (id ⊗ µ) ◦ (δ ⊗ id) = (µ ⊗ id) ◦ (id ⊗ δ) I 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 Frobenius algebra Many definitions over a field k: I algebra A with functional A ! k, kernel without left ideals 9 / 34 I algebra A with nondegenerate A ⊗ A ! k with [ab; c] = [a; bc] I algebra A with comultiplication δ : A ! A ⊗ A satisfying (id ⊗ µ) ◦ (δ ⊗ id) = (µ ⊗ id) ◦ (id ⊗ δ) I 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
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