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The Algebra of DAGs Marcelo Fiore Computer Laboratory University of Cambridge Samson@60 28.V.2013 Joint work with Marco Devesas Campos A Question of Robin Milner A Question of Robin Milner On the generalization from tree structure . . .


  1. The Algebra of DAGs Marcelo Fiore Computer Laboratory University of Cambridge Samson@60 28.V.2013 Joint work with Marco Devesas Campos

  2. A Question of Robin Milner

  3. A Question of Robin Milner On the generalization from tree structure . . . � . . . to dag structure.

  4. Axioms for DAG structure [Gibbons] Problem: Give an algebraic characterisation of the symmetric monoidal category Dag with objects: finite ordinals, and morphisms: finite interfaced dags. 3

  5. 0 0 0 1 1 0 1 2 2 4

  6. Composition: 0 0 0 k 1 1 b 0 1 a 2 2 d l c 0 a b k l 0 1 c d 4-a

  7. � � � � � � � � � The Landscape of Algebraic Structures � Mat N Mat Z Rel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Dag � Dag N � Dag Z Perm Fun POrd � � � � � � � � � � � � � � � � � � � � � � � � � � � � Perm N � Forest N

  8. The Mathematical Setting [Lawvere, MacLane] Symmetric Monoidal Equational Presentations 6

  9. The Mathematical Setting [Lawvere, MacLane] Symmetric Monoidal Equational Presentations Examples: 1. Commutative monoids Operators ∇ : 2 → 1 η : 0 → 1 , Equations ∇ ( x 0 , η ) ≡ x 0 , x 0 ≡ ∇ ( η, x 0 ) ∇ ( x 0 , ∇ ( x 1 , x 2 )) ≡ ∇ ( ∇ ( x 0 , x 1 ) , x 2 ) , ∇ ( x 0 , x 1 ) ≡ ∇ ( x 1 , x 0 ) 6-a

  10. � � � � � � � � � 2. Commutative comonoids Operators ǫ : 1 → 0 , ∆ : 1 → 2 Equations ∆ 1 1 2 2 � � ∆ � � � � � � � id 1 id 1 � � � � � � 1 + ∆ ∆ ∆ � � 1 σ 1,1 � � � � � � � � � � � 1 � 3 � � 1 2 2 ∆ 2 1 + ǫ ǫ + 1 ∆ + 1

  11. PROduct and Permutation categories Definition: A PROP is a symmetric strict monoidal category with underlying monoid structure on objects given by finite ordinals under addition. Examples: 1. Dag 2. The free PROP P [ E ] on a symmetric monoidal equational presentation E . 8

  12. P [ E ] may be constructed syntactically, with morphisms given by equivalence classes of expressions generated by f : ℓ → m , g : m → n id n : n → n f ; g : ℓ → n f 1 : m 1 → n 1 , f 2 : m 2 → n 2 σ m,n : m + n → n + m f 1 + f 2 : m 1 + m 2 → n 1 + n 2 o : n → m an operator o : n → m under the congruence determined by the laws of symmetric strict monoidal categories together with the identities of the equational presentation E .

  13. Algebraic Characterization of DAG Structure Theorem: For D the symmetric monoidal equational presentation of a node together with that of degenerate commutative bialgebras, ∼ P [ D ] = Dag .

  14. � � � � � � � � � Free PROPs 1. The free PROP P [ ∅ ] on the empty equational presentation is the free symmetric strict monoidal category on an object, viz. the category Perm of finite ordinals and permutations . 4 0 1 2 3 0 1 2 3 � � � � � � � � � � � � � � ������������������������ � � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � = � � 4 0 � 1 2 3 � � � � � � � � � � ��������� � ��������� � � � � � � � � � � � � � � � � � � � � � � � � � � 4 0 1 2 3 0 1 2 3 11

  15. 2. The free PROP P [ • ] on the equational presentation of a node • : 1 → 1 is the free symmetric strict monoidal category on the additive monoid of natural numbers 12

  16. � � � � � � � � � 2. The free PROP P [ • ] on the equational presentation of a node • : 1 → 1 is the free symmetric strict monoidal category on the additive monoid of natural numbers, viz. the category Perm N of finite ordinals and N -labelled permutations . 4 0 1 2 3 0 1 2 3 � � � � � � � � � � � � � � � � ��������� � � � � � � m 1 � � � � � � m 1 + n 2 � � � � � � m 0 m 3 � � � � � � � � m 2 � � � � � � � � � � � � m 3 + n 1 � � � � � � � �������������� = � � 4 0 � 1 2 3 � � � � � � � � ������ ��� m 2 + n 3 � � � � � � � n 2 � � � � � ������ n 0 n 3 � � � � m 0 + n 0 � n 1 � � � � � � ��� � � � � 4 0 1 2 3 0 1 2 3 12-a

  17. 3. The free PROP P [ C om M on ] on the equational presentation of commutative monoids is the free cocartesian category on an object, i.e. the category Fun of finite ordinals and functions . 13

  18. � � � 3. The free PROP P [ C om M on ] on the equational presentation of commutative monoids is the free cocartesian category on an object, i.e. the category Fun of finite ordinals and functions . 4 0 1 2 3 0 1 2 3 � � � � ����� � �������������������� ����� � � � � � � � � � � � 4 0 1 2 3 � � � � � ����� � � ����� � = � � � � � � � � � � 3 0 1 2 � � � ����� � ����� � � � � � � 2 0 1 0 1 13-a

  19. 4. The free PROP P [ • + C om C o M on ] on the equational presentation of a node together with that of commutative comonoids is the subcategory Forest of Dag consisting of forests. [Moerdijk, Milner] 14

  20. � � � � � a morphism 4 0 1 2 3 � � ���� � • • � 3 0 1 2 • 3 0 1 2 � � ������� � • � � � � 3 0 1 2 � � � � ���� � • � � � 2 0 1 • 2 0 1 15

  21. � � � � � � � � � � � � a morphism its forest representation 4 0 1 2 3 0 1 2 3 � � ������������� � ���� � � � • • � � � � 3 0 1 2 � � ◦ � • � � � � � � � � 3 0 1 2 � � � � ������� � � � ◦ ◦ ◦ • � � � ������������� � � � � 3 0 1 2 � � � � � � ���� � � • � � � � ◦ ◦ 2 0 1 • 0 1 2 0 1 15-a

  22. � � � � � � � a forest 0 1 2 3 � ���������������� � � � � � � � � ◦ � � � � � � � � � � � � � � � � � ◦ ◦ ◦ � ������������� � � � � � � � � ◦ ◦ 0 1 16

  23. � � � � � � � � � � � � � a layered normal form a forest 4 0 1 2 3 0 1 2 3 � ���������������� � � • � � � 4 0 1 2 3 � � � � ���� � � ◦ � � � � � � � 3 0 1 2 � � � � � • • • � � � � � � ◦ ◦ ◦ 3 0 1 2 � ������������� � ������� � � � � � � � � � � � � 3 0 1 2 � � � � � ���� � ◦ ◦ � � � � 2 0 1 • • 0 1 2 0 1 16-a

  24. 5. The free PROP P [ C om B i A lg ] on the equational presentation of commutative bialgebras is the free category with biproducts on an object, viz. the category Mat N of finite ordinals and N -valued matrices . [MacLane, Pirashvili, Lack] 17

  25. 5. The free PROP P [ C om B i A lg ] on the equational presentation of commutative bialgebras is the free category with biproducts on an object, viz. the category Mat N of finite ordinals and N -valued matrices . [MacLane, Pirashvili, Lack] The equational presentation of commutative bialgebras is that of commutative monoids and commutative comonoids where the comonoid structure is a monoid homomorphism and the comonoid structure is a monoid homomorphism. 17-a

  26. (a) The commutative bialgebra structure turns the symmetric monoidal structure into biproduct structure. 18

  27. (a) The commutative bialgebra structure turns the symmetric monoidal structure into biproduct structure. (b) Every morphism m → n has a unique representation as an m × n matrix with entries in the endo-hom on 1 . 18-a

  28. (a) The commutative bialgebra structure turns the symmetric monoidal structure into biproduct structure. (b) Every morphism m → n has a unique representation as an m × n matrix with entries in the endo-hom on 1 . (c) The endo-hom on 1 is the multiplicative monoid of natural numbers. 0 . n . 0 0 = . n − 1 19-b

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