Coherence modulo and double groupoids Benjamin Dupont Institut - PowerPoint PPT Presentation

Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64.

Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 0 ) 0

Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 0 ) 0 ( C 0 ) 1 � ( C 0 ) 0

� Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 0 ) 0 ( C 0 ) 0 ( C 0 ) 1 � ( C 0 ) 1 ( C 0 ) 0 ( C 0 ) 0

� Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 1 ) 0 � ( C 0 ) 0 ( C 0 ) 0 ( C 0 ) 1 � ( C 0 ) 1 � ( C 0 ) 0 ( C 0 ) 0 ( C 1 ) 0

� � Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 1 ) 0 � ( C 0 ) 0 ( C 0 ) 0 ( C 0 ) 1 � ( C 1 ) 1 ( C 0 ) 1 � ( C 0 ) 0 ( C 0 ) 0 ( C 1 ) 0

� � Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 1 ) 0 � ( C 0 ) 0 ( C 0 ) 0 ( C 0 ) 1 � ( C 1 ) 1 ( C 0 ) 1 � ( C 0 ) 0 ( C 0 ) 0 ( C 1 ) 0 ◮ It gives four related categories C vo := ( C v , C o , ∂ v C ho := ( C h , C o , ∂ h − , 0 , ∂ v + , 0 , ◦ v , i v − , 0 , ∂ h + , 0 , ◦ h , i h 0 ) , 0 ) , C sv := ( C s , C v , ∂ v C sh := ( C s , C h , ∂ h − , 1 , ∂ v + , 1 , ⋄ v , i v − , 1 , ∂ h + , 1 , ⋄ h , i h 1 ) , 1 ) , where C sh is the category C 1 and C vo is the category C 0 .

� � � Double groupoids ◮ We introduce a cubical notion of coherence, in n -categories enriched in double groupoids. ◮ A double category is an internal category ( C 1 , C 0 , ∂ C − , ∂ C + , ◦ C , i C ) in Cat, Ehresmann ’64. ( C 1 ) 0 � ( C 0 ) 0 ( C 0 ) 0 ( C 0 ) 1 � ( C 1 ) 1 ( C 0 ) 1 � ( C 0 ) 0 ( C 0 ) 0 ( C 1 ) 0 ◮ It gives four related categories C vo := ( C v , C o , ∂ v C ho := ( C h , C o , ∂ h − , 0 , ∂ v + , 0 , ◦ v , i v − , 0 , ∂ h + , 0 , ◦ h , i h 0 ) , 0 ) , C sv := ( C s , C v , ∂ v C sh := ( C s , C h , ∂ h − , 1 , ∂ v + , 1 , ⋄ v , i v − , 1 , ∂ h + , 1 , ⋄ h , i h 1 ) , 1 ) , where C sh is the category C 1 and C vo is the category C 0 . ◮ Elements of C o : point cells, elements of C h and C v : horizontal cells and vertical cells. x 1 e f � x 2 x 1 x 2

� � � Double groupoids ◮ Elements of C s are square cells: ∂ h − , 1 ( A ) � · · ∂ v ∂ v − , 1 ( A ) + , 1 ( A ) A � · · ∂ h + , 1 ( A )

� � � � � � � � � � Double groupoids ◮ Elements of C s are square cells: ∂ h i h − , 1 ( A ) � 0 ( x ) � f · · x 1 x 2 x x ∂ v ∂ v , with identities i v i v i v − , 1 ( A ) + , 1 ( A ) i h 1 ( e ) A 0 ( x 1 ) 1 ( f ) 0 ( x 2 ) e e � · � x 2 � y · x 1 y ∂ h + , 1 ( A ) f i h 0 ( y )

� � � � � � � � � � � � � � � � � � � � Double groupoids ◮ Elements of C s are square cells: ∂ h i h − , 1 ( A ) � 0 ( x ) � f · · x 1 x 2 x x ∂ v ∂ v , with identities i v i v i v − , 1 ( A ) + , 1 ( A ) i h 1 ( e ) A 0 ( x 1 ) 1 ( f ) 0 ( x 2 ) e e � · � x 2 � y · x 1 y ∂ h + , 1 ( A ) f i h 0 ( y ) ◮ Compositions f 1 ◦ h f 2 f 1 f 2 � x 3 x 1 x 2 x 1 x 3 A ⋄ v B e 1 e 2 e 3 e 1 e 3 A B � � y 2 � y 3 � y 3 y 1 y 1 g 1 g 2 g 1 ◦ h g 2 i in C v and A , A ′ , B in C s . for all x i , y i , z i in C o , f i in C h , e i , e ′

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Double groupoids ◮ Elements of C s are square cells: ∂ h i h − , 1 ( A ) � 0 ( x ) � f · · x 1 x 2 x x ∂ v ∂ v , with identities i v i v i v − , 1 ( A ) + , 1 ( A ) i h 1 ( e ) A 0 ( x 1 ) 1 ( f ) 0 ( x 2 ) e e � · � x 2 � y · x 1 y ∂ h + , 1 ( A ) f i h 0 ( y ) ◮ Compositions f 1 ◦ h f 2 f 1 f 2 � x 3 x 1 x 2 x 1 x 3 A ⋄ v B e 1 e 2 e 3 e 1 e 3 A B � � y 2 � y 3 � y 3 y 1 y 1 g 1 g 2 g 1 ◦ h g 2 f 1 f 1 x 1 x 2 x 1 x 2 e 1 e 2 A � e 1 ◦ v e ′ e 2 ◦ v e ′ y 1 y 2 A ⋄ h A ′ 1 2 f 2 e ′ e ′ A ′ 1 � 2 � z 2 z 1 � z 2 z 1 f 3 f 3 i in C v and A , A ′ , B in C s . for all x i , y i , z i in C o , f i in C h , e i , e ′

Double groupoids ◮ These compositions satisfy the middle four interchange law:

� � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 x 1 x 2 e 1 � e 2 A � y 2 y 1 g 1

� � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 x 1 x 2 e 1 � e 2 A � y 2 y 1 g 1 ⋄ h g 1 y 1 y 2 e ′ e ′ A ′ 1 � 2 � z 2 z 1 h 1

� � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 x 1 x 2 x 2 x 3 e 1 � e 2 e 2 � e 3 A B � y 2 � y 3 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h g 1 g 2 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ A ′ B ′ 1 � 2 � 2 3 � z 2 � z 3 z 1 z 2 h 1 h 2

� � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 x 1 x 2 x 2 x 3 e 1 � e 2 e 2 � e 3 A B � y 2 � y 3 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = g 1 g 2 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ A ′ B ′ 1 � 2 � 2 3 � z 2 � z 3 z 1 z 2 h 1 h 2

� � � � � � � � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 f 1 f 2 x 1 x 2 x 2 x 3 x 1 x 2 x 2 x 3 ⋄ v e 1 � e 2 e 2 � e 3 e 1 � e 2 e 2 � e 3 A B A B � y 2 � y 3 � y 2 � y 3 y 1 g 1 y 2 g 2 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = g 1 g 2 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ A ′ B ′ 1 � 2 � 2 3 � z 2 � z 3 z 1 z 2 h 1 h 2

� � � � � � � � � � � � � � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 f 1 f 2 x 1 x 2 x 2 x 3 x 1 x 2 x 2 x 3 ⋄ v e 1 � e 2 e 2 � e 3 e 1 � e 2 e 2 � e 3 A B A B � y 2 � y 3 � y 2 � y 3 y 1 g 1 y 2 g 2 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = ⋄ h g 1 g 2 g 1 g 2 y 1 y 2 y 2 y 3 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ e ′ e ′ ⋄ v e ′ e ′ A ′ B ′ A ′ B ′ 1 � 2 � 1 � 2 � 2 3 2 3 � z 2 � z 3 � z 2 � z 3 z 1 z 2 z 1 z 2 h 1 h 2 h 1 h 2

� � � � � � � � � � � � � � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 f 1 f 2 x 1 x 2 x 2 x 3 x 1 x 2 x 2 x 3 ⋄ v e 1 � e 2 e 2 � e 3 e 1 � e 2 e 2 � e 3 A B A B � y 2 � y 3 � y 2 � y 3 y 1 g 1 y 2 g 2 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = ⋄ h g 1 g 2 g 1 g 2 y 1 y 2 y 2 y 3 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ e ′ e ′ ⋄ v e ′ e ′ A ′ B ′ A ′ B ′ 1 � 2 � 1 � 2 � 2 3 2 3 � z 2 � z 3 � z 2 � z 3 z 1 z 2 z 1 z 2 h 1 h 2 h 1 h 2 ◮ Double groupoid: double category in which horizontal, vertical and square cells are invertible.

� � � � � � � � � � � � � � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 f 1 f 2 x 1 x 2 x 2 x 3 x 1 x 2 x 2 x 3 ⋄ v e 1 � e 2 e 2 � e 3 e 1 � e 2 e 2 � e 3 A B A B � y 2 � y 3 � y 2 � y 3 y 1 g 1 y 2 g 2 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = ⋄ h g 1 g 2 g 1 g 2 y 1 y 2 y 2 y 3 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ e ′ e ′ ⋄ v e ′ e ′ A ′ B ′ A ′ B ′ 1 � 2 � 1 � 2 � 2 3 2 3 � z 2 � z 3 � z 2 � z 3 z 1 z 2 z 1 z 2 h 1 h 2 h 1 h 2 ◮ Double groupoid: double category in which horizontal, vertical and square cells are invertible. ◮ n -category enriched in double groupoids: n -category C such that any homset C n ( x , y ) is a double groupoid.

� � � � � � � � � � � � � � � � � � � � � � � � Double groupoids ◮ These compositions satisfy the middle four interchange law: f 1 f 2 f 1 f 2 x 1 x 2 x 2 x 3 x 1 x 2 x 2 x 3 ⋄ v e 1 � e 2 e 2 � e 3 e 1 � e 2 e 2 � e 3 A B A B � y 2 � y 3 � y 2 � y 3 y 1 g 1 y 2 g 2 y 1 g 1 y 2 g 2 ⋄ h ⋄ v ⋄ h = ⋄ h g 1 g 2 g 1 g 2 y 1 y 2 y 2 y 3 y 1 y 2 y 2 y 3 e ′ e ′ e ′ e ′ e ′ e ′ ⋄ v e ′ e ′ A ′ B ′ A ′ B ′ 1 � 2 � 1 � 2 � 2 3 2 3 � z 2 � z 3 � z 2 � z 3 z 1 z 2 z 1 z 2 h 1 h 2 h 1 h 2 ◮ Double groupoid: double category in which horizontal, vertical and square cells are invertible. ◮ n -category enriched in double groupoids: n -category C such that any homset C n ( x , y ) is a double groupoid. ◮ Horizontal ( n + 1 ) -category: category of rewritings, vertical ( n + 1 ) -category: category of modulo rules.

Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories.

Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category.

Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ 0 � � P 0 P 1

� Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ 0 � � 1 P 0 P 1

� Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ 0 � � 1 � � P 0 P 1 P 2

� � Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ P ∗ 0 � � 1 � � 2 P 0 P 1 P 2

� � � � � Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ P ∗ P ∗ P ∗ ( . . . ) � � 0 � � 1 � � 2 � � n − 1 n � � ( . . . ) P 0 P 1 P 2 P n − 1 P n

� � � � � � Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ P ∗ P ∗ P ∗ ( . . . ) � � 0 � � 1 � � 2 � � n − 1 n � � ( . . . ) P 0 P 1 P 2 P n − 1 P n P ⊤ n

� � � � � � Polygraphs ◮ Polygraphs are higher-dimensional generating systems of higher-dimensional globular strict categories. ◮ An n -polygraph generates a free n -category. P ∗ P ∗ P ∗ P ∗ P ∗ ( . . . ) � � 0 � � 1 � � 2 � � n − 1 n � � ( . . . ) P 0 P 1 P 2 P n − 1 P n P ⊤ n ◮ An ( n − 1 ) -category C is presented by an n -polygraph ( P 0 , . . . , P n ) if C ≃ P ∗ n − 1 / ≡ P n

Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of:

�� Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of: ◮ two ( n + 1 ) -polygraphs P v and P h such that P v k = P h k for k ≤ n , P v �� P ∗ P h n + 1 n + 1 n

� � �� � �� � � � Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of: ◮ two ( n + 1 ) -polygraphs P v and P h such that P v k = P h k for k ≤ n , ( P v n + 1 ) ∗ ( P h n + 1 ) ∗ P v P ∗ P h n + 1 n n + 1

� � �� � �� � � � � � � � Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of: ◮ two ( n + 1 ) -polygraphs P v and P h such that P v k = P h k for k ≤ n , ◮ a 2-square extension P s of the pair of ( n + 1 ) -categories (( P v ) ∗ , ( P h ) ∗ ) , that is a set equipped with four maps making Γ a 2-cubical set. P s ( P v n + 1 ) ∗ ( P h n + 1 ) ∗ P v P ∗ P h n + 1 n n + 1

� � �� � � � � � �� � � � Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of: ◮ two ( n + 1 ) -polygraphs P v and P h such that P v k = P h k for k ≤ n , ◮ a 2-square extension P s of the pair of ( n + 1 ) -categories (( P v ) ∗ , ( P h ) ∗ ) , that is a set equipped with four maps making Γ a 2-cubical set. P s ( P v n + 1 ) ∗ ( P h n + 1 ) ∗ P v P ∗ P h n + 1 n n + 1 ◮ A double ( n + 2 , n ) -polygraph is a double n -polygraph in which P s is defined on (( P v ) ⊤ , ( P h ) ⊤ ) .

� � � �� � �� � � � � � � Double ( n + 2 , n ) -polygraphs ◮ A double n -polygraph is a data ( P v , P h , P s ) made of: ◮ two ( n + 1 ) -polygraphs P v and P h such that P v k = P h k for k ≤ n , ◮ a 2-square extension P s of the pair of ( n + 1 ) -categories (( P v ) ∗ , ( P h ) ∗ ) , that is a set equipped with four maps making Γ a 2-cubical set. P s ( P v n + 1 ) ∗ ( P h n + 1 ) ∗ P v P ∗ P h n + 1 n n + 1 ◮ A double ( n + 2 , n ) -polygraph is a double n -polygraph in which P s is defined on (( P v ) ⊤ , ( P h ) ⊤ ) . ◮ A double ( n + 2 , n ) -polygraph ( P v , P h , P s ) generates a free ( n − 1 ) -category enriched in | double groupoids, denoted by ( P v , P h , P s ) = .

� � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ S = ( P h ) ⊤ � · ·

� � � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = there exists a square ( n + 1 ) -cell A in ( P v , P h , P s ) such that ∂ ( A ) = S .

� � � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = there exists a square ( n + 1 ) -cell A in ( P v , P h , P s ) such that ∂ ( A ) = S . ◮ A 2-fold coherent presentation of an n -category C is a double ( n + 2 , n ) -polygraph ( P v , P h , P s ) such that: ◮ the ( n + 1 ) -polygraph P v � P h presents C ; ◮ P s is acyclic

� � � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = there exists a square ( n + 1 ) -cell A in ( P v , P h , P s ) such that ∂ ( A ) = S . ◮ A 2-fold coherent presentation of an n -category C is a double ( n + 2 , n ) -polygraph ( P v , P h , P s ) such that: ◮ the ( n + 1 ) -polygraph P v � P h presents C ; ◮ P s is acyclic ◮ Example: Let E be a convergent ( n + 1 ) -polygraph.

� � � � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = there exists a square ( n + 1 ) -cell A in ( P v , P h , P s ) such that ∂ ( A ) = S . ◮ A 2-fold coherent presentation of an n -category C is a double ( n + 2 , n ) -polygraph ( P v , P h , P s ) such that: ◮ the ( n + 1 ) -polygraph P v � P h presents C ; ◮ P s is acyclic ◮ Example: Let E be a convergent ( n + 1 ) -polygraph. Cd ( E ) := square extension containing = � · · e 1 ⋆ n − 1 e ′ e 2 ⋆ n − 1 e ′ 1 � 2 = � · · for a choice of confluence of any critical branching ( e 1 , e 2 ) of E .

� � � � Acyclicity ◮ A 2-square extension P s of (( P v ) ⊤ , ( P h ) ⊤ ) is acyclic if for any square ( P h ) ⊤ � · · ( P v ) ⊤ ( P v ) ⊤ A S = ( P h ) ⊤ � · · | = there exists a square ( n + 1 ) -cell A in ( P v , P h , P s ) such that ∂ ( A ) = S . ◮ A 2-fold coherent presentation of an n -category C is a double ( n + 2 , n ) -polygraph ( P v , P h , P s ) such that: ◮ the ( n + 1 ) -polygraph P v � P h presents C ; ◮ P s is acyclic ◮ Example: Let E be a convergent ( n + 1 ) -polygraph. Cd ( E ) := square extension containing = � · · e 1 ⋆ n − 1 e ′ e 2 ⋆ n − 1 e ′ 1 � 2 = � · · for a choice of confluence of any critical branching ( e 1 , e 2 ) of E . ◮ From Squier’s theorem, ( E , ∅ , Cd ( E )) is a 2-fold coherent presentation of C .

III. Polygraphs modulo

Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of

Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of ◮ an n -polygraph R of primary rules,

Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of ◮ an n -polygraph R of primary rules, ◮ an n -polygraph E such that E k = R k for k ≤ n − 2 and E n − 1 ⊆ R n − 1 , of modulo rules,

Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of ◮ an n -polygraph R of primary rules, ◮ an n -polygraph E such that E k = R k for k ≤ n − 2 and E n − 1 ⊆ R n − 1 , of modulo rules, ◮ S is a cellular extension of R ∗ n − 1 such that R ⊆ S ⊆ E R E ,

� � � � � Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of ◮ an n -polygraph R of primary rules, ◮ an n -polygraph E such that E k = R k for k ≤ n − 2 and E n − 1 ⊆ R n − 1 , of modulo rules, ◮ S is a cellular extension of R ∗ n − 1 such that R ⊆ S ⊆ E R E , where the cellular extension E R E is defined by γ E R E : E R E → Sph n − 1 ( R ∗ n − 1 ) where E R E is the set of triples ( e , f , e ′ ) in E ⊤ × R ∗ ( 1 ) × E ⊤ such that e u � � v f e ′

� �� � � � Polygraphs modulo A n -polygraph modulo is a data ( R , E , S ) made of ◮ an n -polygraph R of primary rules, ◮ an n -polygraph E such that E k = R k for k ≤ n − 2 and E n − 1 ⊆ R n − 1 , of modulo rules, ◮ S is a cellular extension of R ∗ n − 1 such that R ⊆ S ⊆ E R E , where the cellular extension E R E is defined by γ E R E : E R E → Sph n − 1 ( R ∗ n − 1 ) where E R E is the set of triples ( e , f , e ′ ) in E ⊤ × R ∗ ( 1 ) × E ⊤ such that e u � � v f e ′ and the map γ E R E is defined by γ E R E ( e , f , e ′ ) = ( ∂ − , n − 1 ( e ) , ∂ + , n − 1 ( e ′ )) .

� Branchings and confluence modulo ◮ A branching modulo E of the n -polygraph modulo S is a triple ( f , e , g ) where f and g are in S ∗ n and e is in E ⊤ n , such that: f u ′ u e � g � v ′ v

� Branchings and confluence modulo ◮ A branching modulo E of the n -polygraph modulo S is a triple ( f , e , g ) where f and g are in S ∗ n and e is in E ⊤ n , such that: f u ′ u e � g � v ′ v ◮ It is local if f is in S ∗ ( 1 ) , g is in S ∗ n and e in E ⊤ such that ℓ ( g ) + ℓ ( e ) = 1. n n

� � � Branchings and confluence modulo ◮ A branching modulo E of the n -polygraph modulo S is a triple ( f , e , g ) where f and g are in S ∗ n and e is in E ⊤ n , such that: f u ′ u e � g � v ′ v ◮ It is local if f is in S ∗ ( 1 ) , g is in S ∗ n and e in E ⊤ such that ℓ ( g ) + ℓ ( e ) = 1. n n ◮ It is confluent modulo E if there exists f ′ , g ′ in S ∗ n and e ′ in E ⊤ n : f ′ � w f u ′ u e ′ e � g � v ′ g ′ � w ′ v

� � � Branchings and confluence modulo ◮ A branching modulo E of the n -polygraph modulo S is a triple ( f , e , g ) where f and g are in S ∗ n and e is in E ⊤ n , such that: f u ′ u e � g � v ′ v ◮ It is local if f is in S ∗ ( 1 ) , g is in S ∗ n and e in E ⊤ such that ℓ ( g ) + ℓ ( e ) = 1. n n ◮ It is confluent modulo E if there exists f ′ , g ′ in S ∗ n and e ′ in E ⊤ n : f ′ � w f u ′ u e ′ e � g � v ′ g ′ � w ′ v ◮ Confluence modulo E (resp. local confluence modulo E ): any branching (resp. local branching) of S modulo E is confluent modulo E .

IV. Coherence modulo

Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) .

� � Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) . ◮ A branching modulo E is Γ -confluent modulo E if there exist f ′ , g ′ in S ∗ n , e ′ in E ⊤ n f ′ � w f u ′ u e � e ′ g � v ′ g ′ � w ′ v

� � � Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) . ◮ A branching modulo E is Γ -confluent modulo E if there exist f ′ , g ′ in S ∗ n , e ′ in E ⊤ and a n | = , v : square-cell A in ( E , S , E ⋊ Γ ∪ Peiff ( E , S )) f ′ � f u ′ u w e � e ′ A g � v ′ g ′ � w ′ v

� � � Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) . ◮ A branching modulo E is Γ -confluent modulo E if there exist f ′ , g ′ in S ∗ n , e ′ in E ⊤ and a n | = , v : square-cell A in ( E , S , E ⋊ Γ ∪ Peiff ( E , S )) f ′ � f u ′ u w e � e ′ A g � v ′ g ′ � w ′ v | , v is the free n -category enriched in double categories generated by ( E , S , − ) , in = ◮ ( E , S , − ) which all vertical cells are invertible.

� � � � � Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) . ◮ A branching modulo E is Γ -confluent modulo E if there exist f ′ , g ′ in S ∗ n , e ′ in E ⊤ and a n | = , v : square-cell A in ( E , S , E ⋊ Γ ∪ Peiff ( E , S )) f ′ � f u ′ u w e � e ′ A g � v ′ g ′ � w ′ v | , v is the free n -category enriched in double categories generated by ( E , S , − ) , in = ◮ ( E , S , − ) which all vertical cells are invertible. ◮ Peiff ( E , S ) is the 2-square extension containing the following squares for all e , e ′ ∈ E ⊤ and f ∈ S ∗ . f ⋆ i v � w ⋆ i f � u ′ ⋆ i v w ⋆ i u ′ u ⋆ i v w ⋆ i u u ⋆ i e � u ′ ⋆ i e e ′ ⋆ i u � e ′ ⋆ i u ′ u ⋆ i v ′ f ⋆ i v ′ � u ′ ⋆ i v ′ w ′ ⋆ i u w ′ ⋆ i f � w ′ ⋆ i u ′

� � � � � � � � � Coherent confluence modulo ◮ We consider Γ a 2-square extension of ( E ⊤ , S ∗ ) . ◮ A branching modulo E is Γ -confluent modulo E if there exist f ′ , g ′ in S ∗ n , e ′ in E ⊤ and a n | = , v : square-cell A in ( E , S , E ⋊ Γ ∪ Peiff ( E , S )) f ′ � f u ′ u w e � e ′ A g � v ′ g ′ � w ′ v | , v is the free n -category enriched in double categories generated by ( E , S , − ) , in = ◮ ( E , S , − ) which all vertical cells are invertible. ◮ Peiff ( E , S ) is the 2-square extension containing the following squares for all e , e ′ ∈ E ⊤ and f ∈ S ∗ . f ⋆ i v � w ⋆ i f � u ′ ⋆ i v w ⋆ i u ′ u ⋆ i v w ⋆ i u u ⋆ i e � u ′ ⋆ i e e ′ ⋆ i u � e ′ ⋆ i u ′ u ⋆ i v ′ f ⋆ i v ′ � u ′ ⋆ i v ′ w ′ ⋆ i u w ′ ⋆ i f � w ′ ⋆ i u ′ ◮ E ⋊ Γ is to avoid "redundant" elements in Γ for different squares corresponding to the same branching of S modulo E : f ′ f ′ f f � v ′ � v ′ u v u v e � e ′ and e ⋆ n − 1 e 1 � e ′ u ′ g = e 1 g 1 e 2 � w � w ′ g 1 e 2 � w � w ′ u 1 g ′ g ′

Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E .

Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent:

Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ;

Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ; ◮ S is locally Γ -confluent modulo E ;

� � � � � Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ; ◮ S is locally Γ -confluent modulo E ; ◮ S satisfies properties a ) and b ) : S ∗ � v ′ S ∗ � v ′ S ∗ ( 1 ) � S ∗ ( 1 ) � u v u v E ⊤ ( 1 ) � E ⊤ = A B E ⊤ a) : b) : S ∗ � w ′ R ∗ ( 1 ) � w u ′ � w u S ∗ for any local branching of S modulo E .

� � � � � Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ; ◮ S is locally Γ -confluent modulo E ; ◮ S satisfies properties a ) and b ) : S ∗ � v ′ S ∗ � v ′ S ∗ ( 1 ) � S ∗ ( 1 ) � u v u v E ⊤ ( 1 ) � E ⊤ = A B E ⊤ a) : b) : S ∗ � w ′ R ∗ ( 1 ) � w u ′ � w u S ∗ for any local branching of S modulo E . ◮ S satisfies properties a ) and b ) for any critical branching of S modulo E .

� � � � � Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ; ◮ S is locally Γ -confluent modulo E ; ◮ S satisfies properties a ) and b ) : S ∗ � v ′ S ∗ � v ′ S ∗ ( 1 ) � S ∗ ( 1 ) � u v u v E ⊤ ( 1 ) � E ⊤ = A B E ⊤ a) : b) : S ∗ � w ′ R ∗ ( 1 ) � w u ′ � w u S ∗ for any local branching of S modulo E . ◮ S satisfies properties a ) and b ) for any critical branching of S modulo E . ◮ For S = E R , property b ) is trivially satisfied.

� � � � � Coherent Newman and critical pair lemmas ◮ S is Γ -confluent modulo E (resp. locally Γ -confluent modulo E ) if any of its branching modulo E (resp. local branching modulo E ) is Γ -confluent modulo E . ◮ Theorem. [D.-Malbos ’18] If E R E is terminating, the following assertions are equivalent: ◮ S is Γ -confluent modulo E ; ◮ S is locally Γ -confluent modulo E ; ◮ S satisfies properties a ) and b ) : S ∗ � v ′ S ∗ � v ′ S ∗ ( 1 ) � S ∗ ( 1 ) � u v u v E ⊤ ( 1 ) � E ⊤ = A B E ⊤ a) : b) : R ∗ ( 1 ) � w S ∗ � w ′ u ′ � w u S ∗ for any local branching of S modulo E . ◮ S satisfies properties a ) and b ) for any critical branching of S modulo E . f � ◮ For S = E R , property b ) is trivially satisfied. u v e � v ′