A Solution for the Compositionality Problem of Dinatural Transformations Guy McCusker, Alessio Santamaria 12 th July 2019 Category Theory 2019 Edinburgh, 7- 13 th July 2019
Dinatural transformations F; G : C op × C → D . A dinatural transformation ’ : F → G is a family of morphisms in D ’ = ( ’ A : F ( A; A ) → G ( A; A )) A ∈ C
Dinatural transformations F; G : C op × C → D . A dinatural transformation ’ : F → G is a family of morphisms in D ’ = ( ’ A : F ( A; A ) → G ( A; A )) A ∈ C such that for all f : A → B in C the following commutes: ’ A F ( A; A ) G ( A; A ) F ( f ; 1) G (1 ;f ) F ( B; A ) G ( A; B ) F (1 ;f ) G ( f ; 1) F ( B; B ) G ( B; B ) ’ B
. . . don’t compose ’ : F → G , : G → H dinatural ’ A A F ( A; A ) G ( A; A ) H ( A; A ) F ( f ; 1) G ( f ; 1) G (1 ;f ) H (1 ;f ) F ( B; A ) G ( B; A ) G ( A; B ) H ( A; B ) G (1 ;f ) G ( f ; 1) F (1 ;f ) H ( f ; 1) F ( B; B ) G ( B; B ) H ( B; B ) ’ B B
An extraordinary transformation C cartesian closed category. eval A;B : A × ( A ⇒ B ) → B
An extraordinary transformation C cartesian closed category. eval A;B : A × ( A ⇒ B ) → B eval is natural in B and for all f : A → A ′ the following commutes: 1 × ( f ⇒ 1) A × ( A ′ ⇒ B ) A × ( A ⇒ B ) f × (1 ⇒ 1) eval A;B eval A ′ ;B A ′ × ( A ′ ⇒ B ) B since for all a ∈ A and g : A ′ → B ( g ◦ f )( a ) = g ( f ( a )) .
Extranatural transformations (Eilenberg, Kelly 1966) F : A × B op × B → E , G : A × C op × C → E . An extranatural transformation ’ : F → G is a family of morphisms in E ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A ∈ A ;B ∈ B ;C ∈ C
Extranatural transformations (Eilenberg, Kelly 1966) F : A × B op × B → E , G : A × C op × C → E . An extranatural transformation ’ : F → G is a family of morphisms in E ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A ∈ A ;B ∈ B ;C ∈ C A A ′ , g : B → B B ′ , h : C → C C ′ such that for all f : A → ’ A;B;C F (1 ;g; 1) F ( A; B ′ ; B ) F ( A; B; B ) G ( A; C; C ) F ( A; B; B ) ’ A;B;C F ( f ; 1 ; 1) G ( f ; 1 ; 1) F (1 ; 1 ;g ) ’ A ′ ;B;C ’ A;B ′ ;C F ( A ′ ; B; B ) G ( A ′ ; C; C ) F ( A; B ′ ; B ′ ) G ( A; C; C ) ’ A;B;C F ( A; B; B ) G ( A; C; C ) ’ A;B;C ′ G (1 ; 1 ;h ) G (1 ;h; 1) G ( A; C ′ ; C ′ ) G ( A; C; C ′ )
Extranaturals don’t compose already F : A × B op × B → E , G : A × C op × C → E , H : A × D op × D → E . ’ : F → G , : G → H extranatural transformations. „ « ’ A;B;C A;C;D ◦ ’ = F ( A; B; B ) G ( A; C; C ) H ( A; D; D ) A;B;C;D is not a well-defined extranatural transformation from F to H .
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C !
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! “ ” G , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , , “ ” “ ” F f 1 1 F 1 1 1 , , , , ’ A;B;C F ( A; B; B ) G ( A; C; C ) = F ( f ; 1 ; 1) G ( f ; 1 ; 1) ! A ′ B C A B C ’ A ′ ;B;C F ( A ′ ; B; B ) G ( A ′ ; C; C ) “ ” “ ” G G 1 1 1 1 1 f , , , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , , “ ” “ ” F f F , , , , ’ A;B;C F ( A; B; B ) G ( A; C; C ) = F ( f ; 1 ; 1) G ( f ; 1 ; 1) ! A ′ B C A B C ’ A ′ ;B;C F ( A ′ ; B; B ) G ( A ′ ; C; C ) “ ” “ ” G G f , , , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , , “ ” “ ” g g F F , , , , F (1 ;g; 1) F ( A; B ′ ; B ) F ( A; B; B ) = ’ A;B;C ! F (1 ; 1 ;g ) A B ′ C A B C ’ A;B ′ ;C F ( A; B ′ ; B ′ ) G ( A; C; C ) “ ” “ ” G G , , , ,
A string diagrammatic calculus F : A × B op × B → E , G : A × C op × C → E “ ” F , , ’ = ( ’ A;B;C : F ( A; B; B ) → G ( A; C; C )) A;B;C ! A B C “ ” G , , “ ” “ ” F F , , , , ’ A;B;C F ( A; B; B ) G ( A; C; C ) = ’ A;B;C ′ ! G (1 ; 1 ;h ) C ′ A B A B C G (1 ;h; 1) G ( A; C ′ ; C ′ ) G ( A; C; C ′ ) “ ” “ ” G G h h , , , ,
A string diagrammatic calculus “ ” × ⇒ eval = ( eval A;B : A × ( A ⇒ B ) → B ) A;B ∈ C !
A string diagrammatic calculus “ ” × ⇒ eval = ( eval A;B : A × ( A ⇒ B ) → B ) A;B ∈ C ! g f f “ ” “ ” eval A ′ ;B ′ ⇒ ⇒ × × A ′ × ( A ′ ⇒ B ) B ′ id f × ( id ⇒ g ) = A × ( A ′ ⇒ B ) ! B ′ id × ( f ⇒ id ) g A × ( A ⇒ B ′ ) B g eval A;B
Eilenberg and Kelly’s theorem F ’ G H
Eilenberg and Kelly’s theorem F F ’ ◦ ’ G H H
Eilenberg and Kelly’s theorem F F ’ ◦ ’ G H H Theorem (Eilenberg, Kelly 1966) If the composite graph of ’ and is acyclic, then ◦ ’ is extranatural.
Ramifications in the graphs ∗ C cartesian closed. ( ‹ A : A → A × A ) A ∈ C is a natural transformation f ‹ : id C → × with graph : Naturality of ‹ : = f f ∗ Cf. Kelly, Many-Variable Functorial Calculus I, 1972.
Ramifications in the graphs ∗ C cartesian closed. ( ‹ A : A → A × A ) A ∈ C is a natural transformation f ‹ : id C → × with graph : Naturality of ‹ : = f f Consider ffl A;B = ( ‹ A × id A ⇒ B ) ; ( id A × eval A;B ) : A × ( A ⇒ B ) → A × B . ∗ Cf. Kelly, Many-Variable Functorial Calculus I, 1972.
Ramifications in the graphs ∗ C cartesian closed. ( ‹ A : A → A × A ) A ∈ C is a natural transformation f ‹ : id C → × with graph : Naturality of ‹ : = f f Consider ffl A;B = ( ‹ A × id A ⇒ B ) ; ( id A × eval A;B ) : A × ( A ⇒ B ) → A × B . For f : A → A ′ the following commutes: f f fflA ′ ;B A ′ × ( A ′ ⇒ B ) A ′ × B 1 f × (1 ⇒ 1) = A × ( A ′ ⇒ B ) A ′ × B 1 × ( f ⇒ 1) f × 1 A × ( A ⇒ B ) A × B fflA;B f ffl is natural in B and dinatural in A . ∗ Cf. Kelly, Many-Variable Functorial Calculus I, 1972.
The result F : C ¸ → D , G : C ˛ → D functors, where ¸; ˛ ∈ List { + ; −} , ’ = ( ’ A 1 ;:::;A k ) A 1 ;:::;A k ∈ C : F → G and = ( B 1 ;:::;B l ) B 1 ;:::;B l ∈ C : G → H dinatural transformations with graph Γ( ’ ) and Γ( ) . Theorem If the composition of Γ( ’ ) and Γ( ) is acyclic, then ◦ ’ is again dinatural.
The incidence matrix Say n = number of upper and lower boxes in Γ( ’ ) , m = number of black squares in Γ( ’ ) . The incidence matrix of ’ is the n × m matrix A where 8 − 1 there is an arc from i to j > < A i;j = 1 there is an arc from j to i > 0 otherwise :
The incidence matrix Say n = number of upper and lower boxes in Γ( ’ ) , m = number of black squares in Γ( ’ ) . The incidence matrix of ’ is the n × m matrix A where 8 − 1 there is an arc from i to j > < A i;j = 1 there is an arc from j to i > 0 otherwise : b 1 b 2 b 3 s 1 s 2 b 1 − 1 0 2 3 b 2 1 0 s 1 s 2 6 7 b 3 0 − 1 4 5 b 4 0 1 b 4
The incidence matrix Say n = number of upper and lower boxes in Γ( ’ ) , m = number of black squares in Γ( ’ ) . The incidence matrix of ’ is the n × m matrix A where 8 − 1 there is an arc from i to j > < A i;j = 1 there is an arc from j to i > 0 otherwise : b 1 b 2 b 3 s 1 s 2 f b 1 − 1 0 1 2 3 2 3 b 2 1 0 0 s 1 s 2 6 7 6 7 b 3 0 − 1 0 4 5 4 5 b 4 0 1 0 b 4
The incidence matrix Say n = number of upper and lower boxes in Γ( ’ ) , m = number of black squares in Γ( ’ ) . The incidence matrix of ’ is the n × m matrix A where 8 − 1 there is an arc from i to j > < A i;j = 1 there is an arc from j to i > 0 otherwise : b 1 b 2 b 3 b 1 b 2 b 3 s 1 s 2 f f b 1 − 1 0 1 0 2 3 2 3 2 3 » – b 2 1 0 1 0 1 = 0 + = s 1 s 2 s 1 s 2 6 7 6 7 6 7 b 3 0 − 1 0 0 4 5 4 5 4 5 b 4 0 1 0 0 b 4 b 4
A reachability problem 2 − 1 0 0 0 3 0 1 0 0 6 7 6 7 1 0 − 1 0 6 7 A = 6 7 1 0 0 − 1 6 7 6 7 0 − 1 0 1 4 5 0 0 1 0
A reachability problem 2 − 1 0 0 0 3 0 1 0 0 6 7 6 7 1 0 − 1 0 6 7 A = 6 7 1 0 0 − 1 6 7 6 7 0 − 1 0 1 4 5 0 0 1 0 ◦ ’ · · ( 1 b is a white upper/grey lower box F (1 ;f ) H ( f ; 1) M o ( b ) = 0 otherwise · · ( 1 b is a grey upper/white lower box M d ( b ) = F ( f ; 1) H (1 ;f ) 0 otherwise · · ◦ ’
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