QuotientComprehension Chains Kenta Cho Bart Jacobs k.cho@cs.ru.nl - - PowerPoint PPT Presentation

Quotient–Comprehension Chains

Kenta Cho k.cho@cs.ru.nl Bart Jacobs bart@cs.ru.nl Bas Westerbaan bwesterb@cs.ru.nl Bram Westerbaan awesterb@cs.ru.nl

Radboud University Nijmegen

16 July 2015

A Tree

A Chain of Adjunctions

⊣ ⊣

• ⊣
• ⊣

⊣ ⊣

• ⊣

Quotient

• ⊣

⊣ ⊣

• ⊣

Quotient

• ⊣

Comprehension

Quotient–Comprehension Chains ⊣ ⊣

• ⊣

Quotient

• ⊣

Comprehension

Example: Linear Subspaces

LSub

• Vect

Example: Linear Subspaces

LSub

(V ,S)→V

• Vect

Example: Linear Subspaces

LSub

(V ,S)→V

• Vect

f : (V , S) − → (W , T) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect with f (S) ⊆ T

Example: Linear Subspaces

LSub

(V ,S)→V

• Vect

f : (V , S) − → (W , W ) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

Example: Linear Subspaces

LSub

⊣

• Vect

V →(V ,V )

• f : (V , S) −

→ (W , W ) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

Example: Linear Subspaces

LSub

⊣

• Vect

V →(V ,V )

• f : (V , {0}) −

→ (W , T) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

Example: Linear Subspaces

LSub

⊣ ⊣

• Vect

V →(V ,{0})

• V →(V ,V )
• f : (V , {0}) −

→ (W , T) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

Example: Linear Subspaces

LSub

⊣ ⊣

• Vect
• 1
• f : (V , {0}) −

→ (W , T) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

Example: Linear Subspaces

LSub

⊣ ⊣

• Vect
• 1

Example: Linear Subspaces

LSub

⊣ ⊣

• ⊣

(V ,S)→S

• Vect
• 1

Example: Linear Subspaces

LSub

⊣ ⊣

• ⊣

(V ,S)→S

• Vect
• 1

Example: Linear Subspaces

LSub

⊣ ⊣

• ⊣

(V ,S)→V /S

• ⊣

(V ,S)→S

• Vect
• 1

Example: Linear Subspaces

LSub

⊣ ⊣

• ⊣

(V ,S)→V /S

• ⊣

Comprehension (V ,S)→S

• Vect
• 1

Example: Linear Subspaces

LSub

⊣ ⊣

• ⊣

Quotient (V ,S)→V /S

• ⊣

Comprehension (V ,S)→S

• Vect
• 1

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→V /S

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→V /S

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→S⊥

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→S⊥

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1
• V

quotient of S⊥

S

comprehension of S

V

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→S⊥

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1
• V

quotient of S⊥

• asrtS
• S

comprehension of S

V

Example: Closed Linear Subspaces

CLSub

⊣ ⊣

• ⊣

Quotient (V ,S)→S⊥

• ⊣

Comprehension (V ,S)→S

• Hilb
• 1
• V

quotient of S⊥

• asrtS = projection onto S
• S

comprehension of S

V

Teaser

A → √ B A √ B

Categories with Quotient–Comprehension chain

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)
• 3. (Probabilistic)
• 4. (Quantum)

Boolean Example: Subsets

Pred

(X,S)→S

• Set

Boolean Example: Subsets

Pred

⊣ ⊣

• Set

X→(X,∅)

• X→(X,X)
• f : (X, S) → (Y , T) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in Set with f (S) ⊆ T

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

(X,S)→S

• Set
• 1
• f : (X, S) → (Y , T) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in Set with f (S) ⊆ T

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

✗

• ⊣

(X,S)→S

• Set
• 1
• X → (X, ∅) does not preserve limits

( because (1, ∅) is not final in Pred. )

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

(X,S)→S

• Set+1
• 1
• Set+1 = Kleisli category of (−) + 1

= Sets with partial maps

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

(X,S)→X\S

• ⊣

(X,S)→S

• Set+1
• 1

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• Set+1
• 1

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• Set+1
• 1
• X

quotient of X\S

S

comprehension of S

X

Boolean Example: Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• Set+1
• 1
• X

quotient of X\S

• asrtS :

x→x for x∈S and otherwise undefined

• S

comprehension of S

X

Boolean Example: Clopen Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• Top+1
• 1
• (X, S) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = = clopen S of a topological space X

Boolean Example: Measurable Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• Meas+1
• 1
• (X, S) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = measurable subset S of a measurable space X

Boolean Example: Clopen Subsets

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• CH+1
• 1
• (X, S) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = clopen S of a compact Hausdorff space X

Boolean Example: Projections

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→p⊥A

• ⊣

Comprehension (A ,p)→pA

• (CC∗

MIsU)op

• 1
• (A , p) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = projection p of a commutative unital C ∗-algebra A

Boolean Example: Idempotents

Pred

⊣ ⊣

• ⊣

Quotient (R,e)→e⊥R

• ⊣

Comprehension (R,e)→eR

• (CRngop)+1
• 1
• (R, e) in Pred

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = idempotent p of a commutative unital ring A

Boolean Example: Extensive Category

Pred

⊣ ⊣

• ⊣

Quotient X≡S+S⊥ → S⊥

• ⊣

Comprehension X≡S+S⊥ → S

• E+1
• 1
• (X, S) in Pred

= = = = = = = = = = = = S + S⊥ ≡ X where E is an extensive category with final object, 1

Boolean Example: Boolean Subobjects in a Topos

Pred

⊣ ⊣

• ⊣

Quotient (X,S)→X\S

• ⊣

Comprehension (X,S)→S

• E+1
• 1
• (X, S) in Pred

= = = = = = = = = = = = = = = = = = = S

Boolean subobject

X

where E is a topos

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)
• 3. (Probabilistic)
• 4. (Quantum)

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)
• 4. (Quantum)

Probabilistic Example: K ℓ(D)

Pred

• K

ℓ(D)+1

Probabilistic Example: K ℓ(D)

Pred

• K

ℓ(D≤1)

Probabilistic Example: K ℓ(D)

Pred

• K

ℓ(D≤1) D≤1(X) = { pi |xi : pi ≤ 1 }

Probabilistic Example: K ℓ(D)

Pred

(X,p)→X

• K

ℓ(D≤1) (X, p) in Pred = = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f

Probabilistic Example: K ℓ(D)

Pred

(X,p)→X

• K

ℓ(D≤1) (X, p) in Pred = = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• K

ℓ(D≤1)

• ✗
• (X, p) in Pred

= = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• K

ℓ(D≤1)

• ✗
• (X, p) in Pred

= = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f since ∅ is final in K ℓ(D≤1). . .

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• K

ℓ(D≤1)

• ✗
• (X, p) in Pred

= = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f since ∅ is final in K ℓ(D≤1). . . (∅, q) is final in Pred for some q,

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• K

ℓ(D≤1)

• ✗
• (X, p) in Pred

= = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f since ∅ is final in K ℓ(D≤1). . . (∅, q) is final in Pred for some q, but it is not, since q ◦ f = 0 for all f : X → ∅.

Probabilistic Example: K ℓ(D)

Pred

• K

ℓ(D≤1) (X, p) in Pred = = = = = = = = = = = = p : X → [0, 1] f : (X, p) → (Y , q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in K ℓ(D≤1) with p ≤ q ◦ f + (1 ◦ f )⊥

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{x∈X : p(x)⊥>0}

• ⊣

Comprehension (X,p)→{x∈X : p(x)=1}

• K

ℓ(D≤1)

• 1

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{x∈X : ⌈p(x)⊥⌉=1}

• ⊣

Comprehension (X,p)→{x∈X : p(x)=1}

• K

ℓ(D≤1)

• 1

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{x∈X : ⌈p(x)⊥⌉=1}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(D≤1)

• 1

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{X|⌈p⊥⌉}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(D≤1)

• 1

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{X|⌈p⊥⌉}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(D≤1)

• 1
• X

quotient of p⊥

{X|⌈p⌉}

comprehension of ⌈p⌉ X

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{X|⌈p⊥⌉}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(D≤1)

• 1
• X

quotient of p⊥

• asrtp
• {X|⌈p⌉}

comprehension of ⌈p⌉ X

Probabilistic Example: K ℓ(D)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{X|⌈p⊥⌉}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(D≤1)

• 1
• X

quotient of p⊥

• asrtp : x→p(x) |x
• {X|⌈p⌉}

comprehension of ⌈p⌉ X

Probabilistic Example: K ℓ(G)

Pred

⊣ ⊣

• ⊣

Quotient (X,p)→{X|⌈p⊥⌉}

• ⊣

Comprehension (X,p)→{X|p}

• K

ℓ(G≤1)

• 1

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)
• 4. (Quantum)

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)

K ℓ(D), K ℓ(G), . . .

• 4. (Quantum)

Quantum Example: Operator Algebras

Pred

• (C∗

PsU)op

Quantum Example: Operator Algebras

Pred

(A ,p)→A

• (C∗

PsU)op

(A , p) in Pred = = = = = = = = = = = = p ∈ [0, 1]A

Quantum Example: Operator Algebras

Pred

(A ,p)→A

• (C∗

PsU)op

(A , p) in Pred = = = = = = = = = = = = p ∈ [0, 1]A f : (A , p) → (B, q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : B → A in C∗

PsU , p ≤ f (q) + f (1)⊥

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

?

• ⊣?
• (C∗

PsU)op

• 1
• (A , p) in Pred

= = = = = = = = = = = = p ∈ [0, 1]A f : (A , p) → (B, q) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = f : B → A in C∗

PsU , p ≤ f (q) + f (1)⊥

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣?
• (W∗

PsU)op

• 1

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣?
• (W∗

PsU)op

• 1
• ⌈q⌉ = (least projection above q)

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

?

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• Cauchy–Schwarz inquality for a 2-positive map f : A → B:

f (a∗b)2 ≤ f (a∗a) · f (b∗b)

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• A

Comprehension of ⌈p⌉

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• A

Comprehension of ⌈p⌉

• asrtp

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• A

Comprehension of ⌈p⌉

• asrtp
• a

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• A

Comprehension of ⌈p⌉

• asrtp
• ⌈p⌉a⌈p⌉

a

←

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• A

Comprehension of ⌈p⌉

• asrtp
• √pa√p

⌈p⌉a⌈p⌉

←

a

←

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• ⌈p⌉A ⌈p⌉

u∗au←a

• A
• Compreh. of ⌈p⌉

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (W∗

NCPsU)op

• 1
• A

⌈p⌉A ⌈p⌉

Quotient of p⊥

• ⌈p⌉A ⌈p⌉

u∗au←a

• A
• Compreh. of ⌈p⌉
• √pu∗ a u√p

← a

Quantum Example: Operator Algebras

Pred

⊣ ⊣

• ⊣

Quotient (A ,p)→⌈p⊥⌉A ⌈p⊥⌉

• ⊣

Comprehension (A ,p)→⌊p⌋A ⌊p⌋

• (Fd-C∗

PsU)op

• 1

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)

K ℓ(D), K ℓ(G), . . .

• 4. (Quantum)

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)

K ℓ(D), K ℓ(G), . . .

• 4. (Quantum)

(W∗

NCPsU)op, (Fd-C∗ PsU)op, . . .

Categories with Quotient–Comprehension chain

• 1. Vect, Hilb
• 2. (Boolean)

Set, Top, Meas, CH, (CC∗

MIU)op, CRngop,

any extensive category (with final object, such as a topos)

• 3. (Probabilistic)

K ℓ(D), K ℓ(G), . . .

• 4. (Quantum)

(W∗

NCPsU)op, (Fd-C∗ PsU)op, . . .

• 5. For horrendous counterexamples: OUSop

Questions?