A gentle introduction to 2-dimensional algebra and string diagrams - - PowerPoint PPT Presentation

A gentle introduction to 2-dimensional algebra and string diagrams

Paul-Andr´ e Melli` es CNRS, Universit´ e Paris 7 University Roma 3 October – November 2007

1

The point of view of the logician...

2

Denotational Semantics after Lambek

Cartesian-Closed Categories Intuitionistic Logic λ-calculus

3

Denotational Semantics in the 1990s

Monoidal-Closed Categories Linear Logic Proof-Nets

4

String Diagrams

Categorical Algebra Logic and Language String Diagrams

An algebraic investigation of logic

5

String Diagrams

Categorical Algebra Logic and Language String Diagrams

A logical investigation of algebra

6

String Diagrams

Categorical Algebra Logic and Language String Diagrams

Connections to n-dimensional algebra and physics

7

String Diagrams

Categorical Algebra Logic and Language String Diagrams

Extending the methodology of linear logic to other effects

8

String Diagrams

An idea by Roger Penrose (1970) Further explored by Andr´ e Joyal and Ross Street (1993 – onwards)

9

Monoidal Categories

A monoidal category is a category C equipped with a functor: ⊗ : C × C − → C an object: I and three natural transformations: (A ⊗ B) ⊗ C

α

− → A ⊗ (B ⊗ C) I ⊗ A

λ

− → A A ⊗ I

ρ

− → A satisfying a series of coherence properties.

10

MacLane’s pentagon

(A ⊗ B) ⊗ (C ⊗ D)

α

• ((A ⊗ B) ⊗ C) ⊗ D

α

• α⊗D
• A ⊗ (B ⊗ (C ⊗ D))

(A ⊗ (B ⊗ C)) ⊗ D

α

A ⊗ ((B ⊗ C) ⊗ D)

A⊗α

• The critical pair of the associativity rule.

11

Identity triangle

(A ⊗ I) ⊗ B

α

• ρ⊗B
• A ⊗ (I ⊗ B)

A⊗λ

• A ⊗ B

= A ⊗ B The critical pair of the left and right identity rules.

12

String Diagrams

An idea by Roger Penrose (1970) Further explored by Andr´ e Joyal and Ross Street (1993 – onwards)

13

String Diagrams

A morphism f : A ⊗ B ⊗ C − → D ⊗ E is depicted as:

f A B C D E

14

Composition

The morphism A

f

− → B

g

− → C is depicted as

A A C g ◦ f

=

g f A C B

Vertical composition

15

Tensor product

The morphism (A

f

− → B) ⊗ (C

g

− → D) is depicted as

A ⊗ C B ⊗ D f ⊗ g

=

g f A B C D

Horizontal tensor product

16

Example

f A B D D

f ⊗ idD

17

Example

g f A B C D

(f ⊗ idD) ◦ (idA ⊗ g)

18

Example

g f A B C D

(idB ⊗ g) ◦ (f ⊗ idC)

19

Meaning preserved by deformation

g f A B C D

=

g f A B C D

(f ⊗ idD) ◦ (idA ⊗ g) = (idB ⊗ g) ◦ (f ⊗ idC)

20

Balanced Categories

Axiomatized by Andr´ e Joyal and Ross Street (1993

21

Balanced categories (Joyal, Street 1993)

A balanced category is a monoidal category equipped with Braid maps A ⊗ B

γA,B

− → B ⊗ A

B B A A

Twist maps A θA − → A

A A 22

Braiding

A braiding consists in a family of isomorphisms γA,B : A ⊗ B − → B ⊗ A natural in A and B, which makes the two diagrams commute:

23

Coherence diagram of braiding

A ⊗ I

γ

• ρ
• I ⊗ A

λ

• A

A

24

Coherence diagram of braiding

A ⊗ (B ⊗ C)

α

• A⊗γ
• (A ⊗ B) ⊗ C

γ

C ⊗ (A ⊗ B)

α

• A ⊗ (C ⊗ B)

α

(A ⊗ C) ⊗ B γ⊗B (C ⊗ A) ⊗ B

25

Symmetry

A symmetry is a braiding satisfying that A ⊗ B

γA,B

B ⊗ A

γB,A

A ⊗ B

is equal to the identity.

26

Twist

A twist is a family of isomorphisms A

θA

A

natural in A, and such that θI = idI and the diagram below commutes, for all objects A and B: A ⊗ B

γA,B

• θA⊗B
• B ⊗ A

θB⊗θA

• A ⊗ B

B ⊗ A

γB,A

• 27

Proof nets

An idea by Jean-Yves Girard (1986)

28

Sequent calculus

The two equivalent proofs:

π1 · · · ⊢ A π2 · · · ⊢ B, C ⊢ A ⊗ B, C π3 · · · ⊢ D ⊢ A ⊗ B, C ⊗ D π1 · · · ⊢ A π2 · · · ⊢ B, C π3 · · · ⊢ D ⊢ B, C ⊗ D ⊢ A ⊗ B, C ⊗ D

A permutation equivalence

29

Proof nets

are interpreted by the same proof net:

&

D C B A

π3 π2 π1

A geometric notation

30

Sequentialization by deformation

&

D C B A

π3 π2 π1

π1 · · · ⊢ A π2 · · · ⊢ B, C π3 · · · ⊢ D ⊢ B, C ⊗ D ⊢ A ⊗ B, C ⊗ D

31

Sequentialization by deformation

&

D C B A

π3 π2 π1

π1 · · · ⊢ A π2 · · · ⊢ B, C ⊢ A ⊗ B, C π3 · · · ⊢ D ⊢ A ⊗ B, C ⊗ D

32

Multiplicative proof nets

axiom & & A A* A* A

&

axiom

&

axiom B* A

&

B A* A B A* B* & A B B A

Multiplicative proof nets are string diagrams!

33

Question

It is possible to extend string diagrams with boxes...

34

Functorial boxes

35

Categorical semantics of linear logic (Nick Benton 1994)

A symmetric monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L symmetric monoidal closed

! = L ◦ M

36

Braided linear logic

A balanced monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L balanced monoidal closed

! = L ◦ M

37

Functorial boxes in string diagrams

f F FA FA FB B FB A = Ff

38

Functorial equalities

F g F f f F F g FB FA FA B FA FA C B C FC B A A FA A FA A FC = =

39

Lax monoidal functor

A lax monoidal functor is a functor F : C − → D equipped with mor- phisms m[A,B] : FA ⊗ FB − → F(A ⊗ B) m[−] : I − → FI satisfying a series of coherence relations. A strong monoidal functor is lax monoidal with invertible coercions.

40

The purpose of coercions

FI I F F A2 FA3 FA2 A3 A1 FA1 F(A1 ⊗ A2 ⊗ A3) A1 ⊗ A2 ⊗ A3

m[A1,A2,A3] m[−]

41

Lax monoidal functor

A lax monoidal functor is a box with many inputs - one output.

f F FA1 FAk FB Ak A1 B

F(f) ◦ m[A1,···,Ak] : FA1 ⊗ · · · ⊗ FAk − → FB

42

Functorial equalities (on lax functors)

g F f C FAk A1 B Ak FAj Aj Ai FAi FA1 FC

=

F F g f FA1 FC C B FB Ak A1 FAj FAk Aj Ai FAi B

43

Strong monoidal functors

A strong monoidal functor is a box with many inputs - many outputs

44

Functorial equalities (on strong functors)

g F f B1 FCk FC1 Ck FAi Ai A1 FA1 C1 Bj

=

g F f F FCk FC1 Ck FAi Ai A1 FA1 C1 FBj FB1

45

Functorial equalities (on strong functors)

g F f FD1 FCk Ck C1 FC1 Dl Bj B1 FBk FB1 FAi Ai A1 FA1 D1 FDl

=

F g F f FD1 FCk Ck C1 FC1 Dl Bj B1 FBk FB1 FAi Ai A1 FA1 D1 FDl

46

Natural transformations

A natural transformation θ : F − → G :

C −

→ D satisfies the pictorial equality:

θ f F FA B FB GB A

=

θ f G FA B GA GB A

47

Monoidal natural transformations

A monoidal natural transformation θ : F − → G :

C −

→ D satisfies the pictorial equality:

θ f F FAk FA1 B FB Ak GB A1

=

θ θ f G FAk FA1 GAk GA1 B A1 GB Ak

48

Illustration Decomposing the exponential box of linear logic

49

The categorical semantics of linear logic

A symmetric monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L symmetric monoidal closed

! = L ◦ M

50

Decomposition of the exponential box

! f !B B Ak !Ak !A1 A1

=

L M f MB MAk MA1 B LMAk Ak A1 LMA1 LMB

51

Decomposition of the contraction node

c !A !A !A

=

∆ L MA MA LMA LMA LMA MA

52

Illustration: duplication of the exponential box

L ∆ M L f MB LMB MB LMB MAk MA1 LMB B LMAk Ak A1 LMA1 MB MB

53

Duplication (step 1)

L ∆ M f MB MB LMB MAk MA1 LMB B LMAk Ak A1 LMA1 MB

54

Duplication (step 2)

M L f f ∆ M ∆ MB MA1 MAk MA1 LMAk Ak A1 LMA1 MAk MAk MA1 MB LMB LMB B Ak A1 B

55

Duplication (step 3)

L f L M L f M ∆ ∆ MA1 MB MAk MA1 B LMAk Ak A1 LMA1 MAk MAk MA1 MB LMB LMB B Ak A1 MAk MA1

56

Duplication (step 4)

f M L f L M ∆ ∆ MAk MA1 MB MA1 MAk B LMAk Ak A1 LMA1 LMB MAk MA1 MB LMB LMA1 B Ak A1 LMAk

57

Duplication (step 5)

L L f L M f L M ∆ ∆ MB MAk MA1 B LMAk Ak A1 LMA1 LMB MA1 MB MAk LMB B Ak A1 LMA1 LMAk

58

Five steps instead of one!

Follows faithfully the categorical proof of soundness.

59

Philosophy

Categorical Algebra Logic and Language String Diagrams

60

Traced monoidal categories

61

Trace operator (Joyal - Street - Verity 1996)

A trace in a balanced category C is an operator

A ⊗ U − → B ⊗ U TrU

A,B

A − → B

depicted as feedback in string diagrams:

62

Trace operator

(

)

f f = A A U B B U U TrU

A,B

63

Sliding (naturality in U)

u u f f = A A B B V U U V

64

Tightening (naturality in A, B)

a b a b f f =

65

Vanishing (monoidality in U)

f f = U ⊗ V V U f f = I 66

Superposing

g f f g f g = =

67

Yanking

U U U = = 68

Traces = fixpoints (Hasegawa - Hyland 1997)

In cartesian categories: Fix :

f A U U

→

∆ f A U U

Well-behaved parametric fixpoint operator.

69

Traces = parametric fixpoints (1)

∆ f A U U

70

Traces = parametric fixpoints (2)

∆ ∆ f f A U U

71

Traces = parametric fixpoints (3)

∆ ∆ f f A A U U U

72

Traces = parametric fixpoints (4)

∆ f ∆ f A A U U U

73

Traces = parametric fixpoints (5)

∆ f ∆ f A A A U U

74

Traces = parametric fixpoints (6)

∆ f A U U

=

∆ f ∆ f A U U

75

Illustration Transport of trace

76

Categorical semantics of linear logic (Nick Benton 1994)

A symmetric monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L symmetric monoidal closed

! = L ◦ M

77

Braided linear logic

A balanced monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L balanced monoidal closed

! = L ◦ M

78

Original question

When does a trace in the category L lifts to a trace in the category M ?

M

L

• ⊥

L

M

• Observation: the functor L is usually faithful.

79

Derived question

Characterize when a faithful balanced functor F : C − → D between balanced categories transport a trace in D to a trace in C.

80

Characterization

There exists a trace on C preserved by the functor F ⇐ ⇒ for all objects A, B, U and morphism f : A ⊗ U − → B ⊗ U there exists a morphism g : A − → B such that F(g) = TrFU

FA,FB(m−1 [A,B] ◦ F(f) ◦ m[A,B])

81

Pictorially...

The last equality is depicted as follows:

FU FA FA A FB FB B F f g F =

82

Proof sketch...

First step: define the operator

(

)

f f = A A U B B U U trU

A,B

83

which transports every morphism f to the unique morphism such that

U FU FA FA FB FB F f f = F

Second step: prove that tr satisfies the axioms of a trace operator.

84

Illustration: sliding (1)

We want to show that

u f f u = A A V B U B U V

85

Illustration: sliding (2)

Because the functor F is faithful, this reduces to

u f f u F F = FA FA V FB FB U V U

86

Illustration: sliding (3)

f u F FA FB V U

87

Illustration: sliding (4)

f u F FA FB FV U

88

Illustration: sliding (5)

u F f F FA FV FB U FU

89

Illustration: sliding (6)

u F f F FA FU FB V FV

90

Illustration: sliding (7)

f u F FA FB FU V

91

Illustration: sliding (8)

u f F FA FB U V

92

Examples

• The relational model of linear logic,
• Game semantics (starting from Joyal’s category of Conway games)

Provides well-behaved parametric fixpoints in game semantics

93

Non-Example (Masahito Hasegawa)

The adjunction generated by the powerset monad: category of coalgebras

Set

L

• ⊥

Rel

M

• Kleisli category

The trace of a function in Rel is not a function anymore Here, connections to Ryu Hasegawa and Nicola Gambino’s works.

94

Dualities

95

Duality

A duality between X and Y is a pair of morphisms: X ⊗ Y → I I → Y ⊗ X depicted as

X Y

Y X

96

Duality

satisfying the two equalities:

=

Y Y

=

X X

In that case, one writes X ⊣ Y and says that X is left dual of Y .

97

Tortile category

A balanced category C is called tortile when every object U has a left dual: U∗ ⊣ U.

98

Tortile category (2)

Property: The left duality defines a functor C − → Cop. Every morphism U

f

− → V is transported to the morphism V ∗

f∗

− → U∗ depicted below:

V ∗ U V U ∗ f

99

Tortile category (3)

Property: The left duality functor is strong monoidal C − → Ccoop. Moreover, (A ⊗ B)∗ ∼ = B∗ ⊗ A∗

100

From left to right duality

In a tortile category, the left dual U∗ is also a right dual:

U ∗ U U ∗ U

In fact, the twist θU may be seen as an isomorphism U − → U∗∗.

101

Every tortile category is traced

Property: The left duality functor is strong monoidal C − → Ccoop. (A ⊗ B)∗ ∼ = B∗ ⊗ A∗ Moreover, there is a unique trace.

102

The integer construction

103

Definition of Int(C)

Given a traced monoidal category C, the objects of the category Int(C) are the pairs (A, U) of objects of C, and the morphisms (A, U)

f

(B, V )

• f the category Int(C)

are the morphisms A ⊗ V

B ⊗ U

• f the category C.

Think of an object (A, U) as a “tensorial” fraction A

U.

104

Composition in Int(C)

The composite of two morphisms (A, U)

f

(B, V )

g

(C, W)

is defined by tracing out the object V :

g f A C B U W V 105

Universal property of Int(C)

Theorem (Joyal, Street, Verity 1996) The category Int(C) is tortile. Moreover, the functor

C

− → Int(C) A → (A, I) is full and faithful, and traced monoidal. The functor is also universal, in the appropriate 2-dimensional sense.

106

Hopf and Frobenius objects

107

Hopf objects

A Hopf object H is a monoid and a comonoid satisfying

m d

=

d m m d

=

d m m d

m u

=

u u d e

=

e e

108

Frobenius objects

A Frobenius object F is a monoid and a comonoid satisfying

m d

=

m d

=

m d

Very useful to model networks in concurrency theory.

109

From Hopf to Frobenius objects

Every Hopf object H with a self-duality induces a Frobenius object

d d m m d m

For instance, the set TX of finite multisets on an alphabet X.

110

Associativity (1)

d d d d m m

111

Associativity (2)

d d d m m d d m

112

Associativity (3)

d d d m d m d d

113

Associativity (4)

d d d m

114

Illustration Differential linear logic

115

Differential linear logic

A symmetric monoidal adjunction

M

L

• ⊥

L

M

• M cartesian

L symmetric monoidal closed

+ the category L has biproducts, + the counit ǫ of the comonad ! = L ◦ M has a left inverse ∂ A

∂A

− → !A

ǫA

− → A = A idA − → A with extra axioms induces a differential category in the sense of Blute, Cockett, Seely.

116

The relational model

!A = the free commutative monoid SA computed in the category Set. the counit ǫA : !A → A

• f the comonad !

in the category Rel =

    

the unit ηA : A → SA

• f the monad S

in the category Set

    

• p

All this may be deduced from a distributivity law between S and the powerset monad – following a recipe by Martin Hyland.

117

Relational model

Every function f : A → B in Set induces an adjunction f∗ ⊣ f∗ in Rel (seen as a 2-category) between the opposite relations f∗ : A → B, f∗ : B → A. The function ηA : A → SA is injective, and the induced adjunction ∂A ⊣ ǫA is thus a reflection.

118

Tensorial alchemy

The exponential modality (L, ⊕)

L

− → (M, ×)

M

− → (L, ⊗) transports the biproduct ⊕ into the tensor product ⊗. Moreover, every object A of the category L is a Hopf object in (L, ⊕). Hence: every object !A of the category L is a Hopf object in (L, ⊗).

119

From Hopf to Frobenius objects

For every Hopf object H in a traced monoidal category C, (1) The object (H, I) embedded in Int(C) is a Hopf object, (2) Its dual H∗ = (I, H) in Int(C) is also a Hopf object, (3) The Hopf object ˜ H = H ⊗ H∗ = (H, H) is self-dual, (4) The object ˜ H thus defines a Frobenius object in Int(C). Algebraic reconstruction of the “broadcast zones” in the translation of the π-calculus in differential interaction nets by Ehrhard and Laurent.

120

Thank you!

121

Interlude on knots.

122

Knots: a bestiary

Two knots equal by deformation

∼

124

The three Reidemeister moves

∼ ∼ ∼ ∼ ∼ ∼

125

Two knots people believed different in 1885...

∼

but Kenneth Perko discovers a series of moves in 1974 !

126

How does one show that two knots are different?

∼

left trefoil right trefoil

127

Answer: knot invariants

John Horton Conway Vaughan Jones

128

Step 1: orient the knot

→

129

Step 2: decompose the knot in generators

X+ X− double arrow U

130

Step 2: decompose the knot in generators

→

131

Step 2: decompose the knot in generators

→

132

Step 3: replace each generator by a polynomial

→

polynôme polynôme polynôme polynôme polynôme polynôme polynôme

133

Step 4: compose the polynomial 2-dimensionally

polynôme polynôme polynôme polynôme polynôme polynôme polynôme

→

polynôme polynôme polynôme polynôme polynôme

134

Step 4: compose the polynomials 2-dimensionally

polynôme polynôme polynôme polynôme polynôme polynôme polynôme

→

polynôme polynôme polynôme polynôme

135

Step 4: compose the polynomials 2-dimensionally

polynôme polynôme polynôme polynôme polynôme polynôme polynôme

→

polynôme polynôme polynôme

136

Step 4: compose the polynomials 2-dimensionally

polynôme polynôme polynôme polynôme polynôme polynôme polynôme

→

polynôme

137

Important: the translation is modular

Same principles as in semantics for proofs and programs.

138

Step 5: check that the polynomial defines an invariant

= =

It is sufficient to test the invariant on the Reidemeister moves!

139

Recipe to compute the Conway-Jones invariant

x =

1 x

+ y

[X+] [X−] [double arrow]

140

Recette pour calculer l’invariant de Conway-Jones

→

1

141

Application: invariant of right trefoil

x =

1 x

+ y

[X+] [X−] [double arrow]

142

Application: invariant of right trefoil

x =

1 x

+ y

[X+] [X−] [double arrow]

143

Application: invariant of right trefoil

x =

1 x

+ y

[X+] [X−] [double arrow]

144

Intermediate calculus: the intertwined loops

∼

Par simple rotation

145

Intermediate calculus: the intertwined loops

x =

1 x

+ y

[X+] [X−] [double arrow]

146

Intermediate calculus: the intertwined loops

x =

1 x

+ y

[X+] [X−] [double arrow]

147

Intermediate calculus: the double circle

x =

1 x

+ y

[X+] [X−] [double arrow]

148

First result: invariant of the double circle

=

x y − 1 xy

149

Back to the intertwined loops

x =

1 x

+ y

[X+] [X−] [double arrow]

150

Back to the intertwined loops

x =

1 y − 1 x2y +

y

[X+] [X−] [double arrow]

151

Second result: invariant of the intertwined loops

=

1 xy − 1 x3y + y x

152

Back to the right trefoil

x =

1 x

+ y

[X+] [X−] [double arrow]

153

Back to the right trefoil

x =

1 x

+

1 x + 1 x3 + y2 x

[X+] [X−] [double arrow]

154

Result of the calculus: the invariant of the right trefoil

=

2 x2 + 1 x4 + y2 x2

155

Let us compare it to the invariant of the left trefoil

=

2 x2 + 1 x4 + y2 x2

= 2x2 − x4 + x2y2

The left and the right trefoil are not equivalent.

156