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

Quotient–Comprehension Chains Kenta Cho Bart Jacobs k.cho@cs.ru.nl bart@cs.ru.nl Bas Westerbaan Bram Westerbaan bwesterb@cs.ru.nl awesterb@cs.ru.nl Radboud University Nijmegen 16 July 2015

A Tree

A Chain of Adjunctions

� � � � � ⊣ ⊣ ⊣ ⊣

� � � � � Quotient ⊣ ⊣ ⊣ ⊣

� � � � � Comprehension Quotient ⊣ ⊣ ⊣ ⊣

� � � � � Quotient–Comprehension Chains Comprehension Quotient ⊣ ⊣ ⊣ ⊣

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

� � Example: Linear Subspaces LSub V �→ ( V , V ) ⊣ Vect f : ( V , { 0 } ) − → ( W , T ) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

� � � Example: Linear Subspaces LSub V �→ ( V , { 0 } ) V �→ ( V , V ) ⊣ ⊣ Vect f : ( V , { 0 } ) − → ( W , T ) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

� � � Example: Linear Subspaces LSub ⊣ ⊣ 0 1 Vect f : ( V , { 0 } ) − → ( W , T ) in LSub = = = = = = = = = = = = = = = = = = = = = = = = = = f : V → W in Vect

� � � Example: Linear Subspaces LSub ⊣ ⊣ 0 1 Vect

� � � � Example: Linear Subspaces LSub ⊣ ⊣ ⊣ ( V , S ) �→ S 0 1 Vect

� � � � Example: Linear Subspaces LSub ⊣ ⊣ ⊣ ( V , S ) �→ S 0 1 Vect

� � � � � Example: Linear Subspaces LSub ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ V / S ( V , S ) �→ S 0 1 Vect

� � � � � Example: Linear Subspaces LSub Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ V / S ( V , S ) �→ S 0 1 Vect

� � � � � Example: Linear Subspaces LSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ V / S ( V , S ) �→ S 0 1 Vect

� � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ V / S ( V , S ) �→ S 0 1 Hilb

� � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ V / S ( V , S ) �→ S 0 1 Hilb

� � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ S ⊥ ( V , S ) �→ S 0 1 Hilb

� � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ S ⊥ ( V , S ) �→ S 0 1 Hilb quotient of S ⊥ comprehension of S � S � V V

� � � � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ S ⊥ ( V , S ) �→ S 0 1 Hilb quotient of S ⊥ comprehension of S � V V S asrt S

� � � � � � � Example: Closed Linear Subspaces CLSub Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( V , S ) �→ S ⊥ ( V , S ) �→ S 0 1 Hilb quotient of S ⊥ comprehension of S � V V S asrt S = projection onto S

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 X �→ ( X , ∅ ) X �→ ( X , X ) ⊣ ⊣ Set f : ( X , S ) → ( Y , T ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in Set with f ( S ) ⊆ T

� � � � Boolean Example: Subsets Pred ⊣ ⊣ ⊣ ( X , S ) �→ S 0 1 Set f : ( X , S ) → ( Y , T ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = f : X → Y in Set with f ( S ) ⊆ T

� � � � � Boolean Example: Subsets Pred ✗ ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ S 0 1 Set X �→ ( X , ∅ ) does not preserve limits ( because (1 , ∅ ) is not final in Pred . )

� � � � Boolean Example: Subsets Pred ⊣ ⊣ ⊣ ( X , S ) �→ S 0 1 Set +1 Set +1 = Kleisli category of ( − ) + 1 = Sets with partial maps

� � � � � Boolean Example: Subsets Pred ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Set +1

� � � � � Boolean Example: Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Set +1

� � � � � Boolean Example: Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Set +1 quotient of X \ S comprehension of S � S � X X

� � � � � � � Boolean Example: Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Set +1 quotient of X \ S comprehension of S � X X S asrt S : x �→ x for x ∈ S and otherwise undefined

� � � � � Boolean Example: Clopen Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Top +1 ( X , S ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = clopen S of a topological space X

� � � � � Boolean Example: Measurable Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 Meas +1 ( X , S ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = measurable subset S of a measurable space X

� � � � � Boolean Example: Clopen Subsets Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 CH +1 ( X , S ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = clopen S of a compact Hausdorff space X

� � � � � Boolean Example: Projections Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( A , p ) �→ p ⊥ A ( A , p ) �→ p A 0 1 ( CC ∗ MIsU ) op ( A , p ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = projection p of a commutative unital C ∗ -algebra A

� � � � � Boolean Example: Idempotents Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ ( R , e ) �→ e ⊥ R ( R , e ) �→ eR 0 1 ( CRng op ) +1 ( R , e ) in Pred = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = idempotent p of a commutative unital ring A

� � � � � Boolean Example: Extensive Category Pred Quotient Comprehension ⊣ ⊣ ⊣ ⊣ X ≡ S + S ⊥ �→ S ⊥ X ≡ S + S ⊥ �→ S 0 1 E +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 Comprehension ⊣ ⊣ ⊣ ⊣ ( X , S ) �→ X \ S ( X , S ) �→ S 0 1 E +1 ( X , S ) in Pred = = = = = = = = = = = = = = = = = = = � X S Boolean subobject 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 Set , Top , Meas , CH , ( CC ∗ MIU ) op , CRng op , 2. (Boolean) 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 ) = { � p i | x i � : � p i ≤ 1 }

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