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 † Terminology: adjoints in Hilbert spaces hf(x) | yiY = hx | 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 †) adjoint, and up to scalar any adjunction of this form

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 hf(x) | yiY = hx | 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 †) 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 † idempotent projection f = f † ◦ f functor dagger functor F (f †) = F (f)† † † natural transform natural transformation (α )X = (αX ) monoidal structure monoidal dagger structure (f ⊗ g)† = f † ⊗ g†

5 / 19 isn’t this trivially trivial?

Way of the dagger

Category theory Dagger category theory

5 / 19 Way of the dagger

Category theory Dagger category theory

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 † + laws =⇒ Q† + 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 † + laws =⇒ Q† + 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 † + laws =⇒ Q† + laws for dagger categories

6 / 19 Kl(GF ) D FEM(GF )

F G

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

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

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 µ† = T µ ◦ µ†T

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 µ† = T µ ◦ µ†T

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 µ† = T µ ◦ µ†T

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

† T (f †) 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

† T (f †) 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)† T (A) T 2(A) µ† 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)† T (A) T 2(A) µ† 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)† T (A) T 2(A) µ† 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. † I EM-algebra (A, a) is FEM iﬀ a is morphism (A, a) → (T A, µA) a I (A, a) ∈ Im(J) associative =⇒ T A, µA) → (A, a) ∈ Im(J) =⇒ a† ∈ Im(J) =⇒ (A, a) ∈ FEM(GF )

Dagger monads Theorem If F,G are dagger adjoint, there are unique dagger functors with

K J Kl(GF ) D FEM(GF )

F G

J is full, K is full and faithful, and JK is the canonical inclusion

11 / 19 a I (A, a) ∈ Im(J) associative =⇒ T A, µA) → (A, a) ∈ Im(J) =⇒ a† ∈ Im(J) =⇒ (A, a) ∈ FEM(GF )

Dagger monads Theorem If F,G are dagger adjoint, there are unique dagger functors with

K J Kl(GF ) D FEM(GF )

F G

J is full, K is full and faithful, and JK is the canonical inclusion Proof. † I EM-algebra (A, a) is FEM iﬀ a is morphism (A, a) → (T A, µA)

11 / 19 Dagger monads Theorem If F,G are dagger adjoint, there are unique dagger functors with

K J Kl(GF ) D FEM(GF )

F G

J is full, K is full and faithful, and JK is the canonical inclusion Proof. † I EM-algebra (A, a) is FEM iﬀ a is morphism (A, a) → (T A, µA) a I (A, a) ∈ Im(J) associative =⇒ T A, µA) → (A, a) ∈ Im(J) =⇒ a† ∈ Im(J) =⇒ (A, a) ∈ FEM(GF )

11 / 19 I Dagger monad is strong when strength is unitary

I Frobenius monoids in C ' strong dagger monads on C M 7→ − ⊗ M T (I) ← T [ I [Z, FHilb] → [N, FHilb] has dagger adjoint f 7→ Im(f) but induced monad decreases dimension so not strong

I If T commutative, then Kl(T ) dagger symmetric monoidal

Strength

I Monad T is strong when coherent natural A ⊗ T (B) → T (A ⊗ B)

I monoids in C ' monads on C M 7→ − ⊗ M T (I) ← T [

12 / 19 I [Z, FHilb] → [N, FHilb] has dagger adjoint f 7→ Im(f) but induced monad decreases dimension so not strong

I If T commutative, then Kl(T ) dagger symmetric monoidal

Strength

I Monad T is strong when coherent natural A ⊗ T (B) → T (A ⊗ B)

I monoids in C ' monads on C M 7→ − ⊗ M T (I) ← T [

I Dagger monad is strong when strength is unitary

I Frobenius monoids in C ' strong dagger monads on C M 7→ − ⊗ M T (I) ← T [

12 / 19 I If T commutative, then Kl(T ) dagger symmetric monoidal

Strength

I Monad T is strong when coherent natural A ⊗ T (B) → T (A ⊗ B)

I monoids in C ' monads on C M 7→ − ⊗ M T (I) ← T [

I Dagger monad is strong when strength is unitary

I Frobenius monoids in C ' strong dagger monads on C M 7→ − ⊗ M T (I) ← T [ I [Z, FHilb] → [N, FHilb] has dagger adjoint f 7→ Im(f) but induced monad decreases dimension so not strong

12 / 19 Strength

I Monad T is strong when coherent natural A ⊗ T (B) → T (A ⊗ B)

I monoids in C ' monads on C M 7→ − ⊗ M T (I) ← T [

I Dagger monad is strong when strength is unitary

I Frobenius monoids in C ' strong dagger monads on C M 7→ − ⊗ M T (I) ← T [ I [Z, FHilb] → [N, FHilb] has dagger adjoint f 7→ Im(f) but induced monad decreases dimension so not strong

I If T commutative, then Kl(T ) dagger symmetric monoidal

12 / 19 Dagger limits

Should:

I be unique up to unique unitary

I be deﬁned canonically (without e.g. enrichment)

I generalize dagger biproducts and dagger equalisers

I connect to dagger adjunctions and dagger Kan extensions

13 / 19 I Two limits are unitarily iso iﬀ

A L

M B

commutes for all A, B

I Right notion of dagger limit means ﬁxing maps A → L → B.

Unique up to unitary

f I Two limits (L, lA), (M, mA) of same diagram are iso L → M. Now f −1 is iso of limits M → L. But f † is iso of colimits.

14 / 19 I Right notion of dagger limit means ﬁxing maps A → L → B.

Unique up to unitary

f I Two limits (L, lA), (M, mA) of same diagram are iso L → M. Now f −1 is iso of limits M → L. But f † is iso of colimits.

I Two limits are unitarily iso iﬀ

A L

M B

commutes for all A, B

14 / 19 Unique up to unitary

f I Two limits (L, lA), (M, mA) of same diagram are iso L → M. Now f −1 is iso of limits M → L. But f † is iso of colimits.

I Two limits are unitarily iso iﬀ

A L

M B

commutes for all A, B

I Right notion of dagger limit means ﬁxing maps A → L → B.

14 / 19 ⇐⇒ dagger D : J → C have compatible dagger Kan extension along J → 1 with ε ◦ ε† idempotent

Theorem C has all J-shaped limits ⇐⇒ ∆: C → [J, C] has dagger adjoint and ε ◦ ε† idempotent

Proof. † lJ ◦ lJ is largest projection compatible with D

Dagger-shaped limits

Deﬁnition The dagger limit of dagger functor D : J → C is a limit (L, lJ ) with † I each lJ ◦ lJ is projection; I lK ◦ lJ = 0 when J(J, K) = ∅.

15 / 19 ⇐⇒ dagger D : J → C have compatible dagger Kan extension along J → 1 with ε ◦ ε† idempotent Proof. † lJ ◦ lJ is largest projection compatible with D

Dagger-shaped limits

Deﬁnition The dagger limit of dagger functor D : J → C is a limit (L, lJ ) with † I each lJ ◦ lJ is projection; I lK ◦ lJ = 0 when J(J, K) = ∅.

Theorem C has all J-shaped limits ⇐⇒ ∆: C → [J, C] has dagger adjoint and ε ◦ ε† idempotent

15 / 19 ⇐⇒ dagger D : J → C have compatible dagger Kan extension along J → 1 with ε ◦ ε† idempotent

Dagger-shaped limits

Deﬁnition The dagger limit of dagger functor D : J → C is a limit (L, lJ ) with † I each lJ ◦ lJ is projection; I lK ◦ lJ = 0 when J(J, K) = ∅.

Theorem C has all J-shaped limits ⇐⇒ ∆: C → [J, C] has dagger adjoint and ε ◦ ε† idempotent

Proof. † lJ ◦ lJ is largest projection compatible with D

15 / 19 Dagger-shaped limits

Deﬁnition The dagger limit of dagger functor D : J → C is a limit (L, lJ ) with † I each lJ ◦ lJ is projection; I lK ◦ lJ = 0 when J(J, K) = ∅.

Theorem C has all J-shaped limits ⇐⇒ ∆: C → [J, C] has dagger adjoint and ε ◦ ε† idempotent ⇐⇒ dagger D : J → C have compatible dagger Kan extension along J → 1 with ε ◦ ε† idempotent Proof. † lJ ◦ lJ is largest projection compatible with D

15 / 19 I Dagger stabiliser: J = Free · ·

I Dagger projection: inﬁmum of projections pj : J → J splits

I C has dagger limits of dagger shapes with κ components ⇐⇒ C has dagger limits of

I dagger products of size κ I dagger stabilisers I dagger projections

Constructing dagger-shaped limits

pJ pK † I Dagger product: product J ←− J × K −→ K with pK pJ = δJK

e † I Dagger equaliser: equaliser E J K with e e = id

16 / 19 I Dagger projection: inﬁmum of projections pj : J → J splits

I C has dagger limits of dagger shapes with κ components ⇐⇒ C has dagger limits of

I dagger products of size κ I dagger stabilisers I dagger projections

Constructing dagger-shaped limits

pJ pK † I Dagger product: product J ←− J × K −→ K with pK pJ = δJK

e † I Dagger equaliser: equaliser E J K with e e = id

I Dagger stabiliser: J = Free · ·

16 / 19 I C has dagger limits of dagger shapes with κ components ⇐⇒ C has dagger limits of

I dagger products of size κ I dagger stabilisers I dagger projections

Constructing dagger-shaped limits

pJ pK † I Dagger product: product J ←− J × K −→ K with pK pJ = δJK

e † I Dagger equaliser: equaliser E J K with e e = id

I Dagger stabiliser: J = Free · ·

I Dagger projection: inﬁmum of projections pj : J → J splits

16 / 19 Constructing dagger-shaped limits

pJ pK † I Dagger product: product J ←− J × K −→ K with pK pJ = δJK

e † I Dagger equaliser: equaliser E J K with e e = id

I Dagger stabiliser: J = Free · ·

I Dagger projection: inﬁmum of projections pj : J → J splits

I C has dagger limits of dagger shapes with κ components ⇐⇒ C has dagger limits of

I dagger products of size κ I dagger stabilisers I dagger projections

16 / 19 2 2 2 2 ··· C C C ···

Non-dagger shapes?

What to do with loops?

2 C C 2 1 4 C

17 / 19 Non-dagger shapes?

What to do with loops?

2 C C 2 1 4 C

2 2 2 2 ··· C C C ···

17 / 19 ... but they ain’t all that bad

I Dagger equivalence is equivalence in DagCat unitary (co)unit

F C D I If C ∈ DagCat, when does equivalence G in Cat lift to dagger equivalence? Clearly need η and Gε unitary.

Theorem: this is suﬃcient.

I Theorem: If there is unitary GF A → A for each A, can replace F,G with isomorphic functors that lift to dagger equivalence.

Daggers are evil

I No dagger on FVect respects forgetful FHilb → FVect. Proof: equip vector space with two inner products; then v 7→ v not unitary but maps to identity

18 / 19 ... but they ain’t all that bad

Theorem: this is suﬃcient.

I Theorem: If there is unitary GF A → A for each A, can replace F,G with isomorphic functors that lift to dagger equivalence.

Daggers are evil

I No dagger on FVect respects forgetful FHilb → FVect. Proof: equip vector space with two inner products; then v 7→ v not unitary but maps to identity

I Dagger equivalence is equivalence in DagCat unitary (co)unit

F C D I If C ∈ DagCat, when does equivalence G in Cat lift to dagger equivalence? Clearly need η and Gε unitary.

18 / 19 I Theorem: If there is unitary GF A → A for each A, can replace F,G with isomorphic functors that lift to dagger equivalence.

Daggers are evil ... but they ain’t all that bad

I No dagger on FVect respects forgetful FHilb → FVect. Proof: equip vector space with two inner products; then v 7→ v not unitary but maps to identity

I Dagger equivalence is equivalence in DagCat unitary (co)unit

F C D I If C ∈ DagCat, when does equivalence G in Cat lift to dagger equivalence? Clearly need η and Gε unitary.

Theorem: this is suﬃcient.

18 / 19 Daggers are evil ... but they ain’t all that bad

I No dagger on FVect respects forgetful FHilb → FVect. Proof: equip vector space with two inner products; then v 7→ v not unitary but maps to identity

I Dagger equivalence is equivalence in DagCat unitary (co)unit

F C D I If C ∈ DagCat, when does equivalence G in Cat lift to dagger equivalence? Clearly need η and Gε unitary.

Theorem: this is suﬃcient.

I Theorem: If there is unitary GF A → A for each A, can replace F,G with isomorphic functors that lift to dagger equivalence.

18 / 19 Conclusion

I DagCat is not just a 2-category so dagger category theory nontrivial

I Dagger monads = monad + dagger functor + Frobenius law

I Dagger-shaped limits = limit + dagger + idempotent Dagger limits = ?

I Dagger categories can’t be that evil

19 / 19