Outline
Motivation
Natural Numbers Objects
Dialectica Categories
NNOs
Conclusions

Natural Number Objects in Dialectica Ca...
Outline
Motivation
Natural Numbers Objects
Dialectica Categories
NNOs
Conclusions

Motivation
Natural Numbers Objects
Dial...
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Natural Numbers Objects
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Linear Logic perspective into recursion...
Outline
Motivation
Natural Numbers Objects
Dialectica Categories
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Linear Logic perspective into recursion...
Outline
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Natural Numbers Objects
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Linear Logic perspective into recursion...
Outline
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Natural Numbers Objects
Dialectica Categories
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Linear Recursion and Iteration?

Iterat...
Outline
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Natural Numbers Objects
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Linear Recursion and Iteration?

Iterat...
Outline
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Natural Numbers Objects
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Linear Recursion and Iteration?

Iterat...
Outline
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Natural Numbers Objects
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Categorical Models of PCF?

Cartesian c...
Outline
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Natural Numbers Objects
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Categorical Models of PCF?

Cartesian c...
Outline
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Natural Numbers Objects
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NNOs
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Categorical Models of PCF?

Cartesian c...
Outline
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Natural Numbers Objects
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NNOs
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Categorical Models of PCF?

Cartesian c...
Outline
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Natural Numbers Objects
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What are Natural Numbers Objects?

Lawv...
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Natural Numbers Objects
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What are Natural Numbers Objects?

Lawv...
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Natural Numbers Objects
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What are Natural Numbers Objects?

Lawv...
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Natural Numbers Objects
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What are Natural Numbers Objects?

Lawv...
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Natural Numbers Objects
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What are Natural Numbers Objects?
A NNO...
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Natural Numbers Objects in Sets
A NNO i...
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Natural Numbers Objects in Sets
Given f...
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Natural Numbers Objects in a SMCC?
Par´...
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Dialectica Categories

Conceived as an ...
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Dialectica Categories
A morphism from A...
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Tensor in Dialectica Categories
Let A =...
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Symmetric Monoidal Closed Dialectica Ca...
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Other structure of Dialectica Categorie...
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Cartesian structure of Dialectica Categ...
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NNOs in Dialectica Categories?

To inve...
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NNOs in Dialectica Categories?
Reading ...
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NNO using cartesian structure
∅
∅

 
 
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NNO using cartesian structure
Propositi...
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NNO using cartesian structure
This triv...
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NNO using monoidal structure
[Burroni] ...
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NNO using monoidal structure
If a categ...
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NNO using monoidal structure
B
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A

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NNO using monoidal structure
Simplifyin...
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NNO using monoidal structure
Propositio...
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NNO using monoidal structure
The morphi...
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NNO using monoidal structure

Fact 2. I...
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NNO using monoidal structure
The fact t...
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NNO using monoidal structure
Back to th...
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NNO using monoidal structure
The
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NNO using monoidal structure

Items 2 a...
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NNO using monoidal structure
If m = 0 w...
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NNO using monoidal structure
If m is no...
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Summing up
We obtained a degenerate wea...
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Summing up
Natural number algebras 1 → ...
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Some references
Sandra Alves, Maribel F...
Outline
Motivation
Natural Numbers Objects
Dialectica Categories
NNOs
Conclusions

Thanks!

Valeria de Paiva, Charles Morg...
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Natural Number Objects in Dialectica Categories

  1. 1. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Natural Number Objects in Dialectica Categories Valeria de Paiva Charles Morgan Samuel Gomes da Silva September 2, 2013 Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  2. 2. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  3. 3. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Logic perspective into recursion and iteration? Linear Logic can be seen as magnifying lens to understand logic Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  4. 4. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Logic perspective into recursion and iteration? Linear Logic can be seen as magnifying lens to understand logic Decomposing implication via Girard’s translation A → B :=!A−◦B gives new insights on computational phenomena Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  5. 5. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Logic perspective into recursion and iteration? Linear Logic can be seen as magnifying lens to understand logic Decomposing implication via Girard’s translation A → B :=!A−◦B gives new insights on computational phenomena Want to use linear perspective for iteration and recursion Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  6. 6. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Recursion and Iteration? Iteration and recursion in intuitionistic logic done via PCF Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  7. 7. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Recursion and Iteration? Iteration and recursion in intuitionistic logic done via PCF the mother of all programming languages Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  8. 8. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Linear Recursion and Iteration? Iteration and recursion in intuitionistic logic done via PCF the mother of all programming languages and its denotational models Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  9. 9. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Categorical Models of PCF? Cartesian closed category Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  10. 10. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Categorical Models of PCF? Cartesian closed category with booleans, NNO and fixpoints Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  11. 11. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Categorical Models of PCF? Cartesian closed category with booleans, NNO and fixpoints Want to start with Linear PCF and linear natural number objects... Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  12. 12. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Categorical Models of PCF? Cartesian closed category with booleans, NNO and fixpoints Want to start with Linear PCF and linear natural number objects... This talk: linear natural number objects in a specific model of Linear Logic, Dialectica categories Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  13. 13. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions What are Natural Numbers Objects? Lawvere’s way of modelling the natural numbers with Peano’s Induction Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  14. 14. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions What are Natural Numbers Objects? Lawvere’s way of modelling the natural numbers with Peano’s Induction CCC with NNO ⇒ all primitive recursive functions. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  15. 15. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions What are Natural Numbers Objects? Lawvere’s way of modelling the natural numbers with Peano’s Induction CCC with NNO ⇒ all primitive recursive functions. A Natural Numbers Object (or NNO) is an object in a category equipped with structure giving it properties similar to those of the set of natural numbers N in the category Sets. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  16. 16. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions What are Natural Numbers Objects? Lawvere’s way of modelling the natural numbers with Peano’s Induction CCC with NNO ⇒ all primitive recursive functions. A Natural Numbers Object (or NNO) is an object in a category equipped with structure giving it properties similar to those of the set of natural numbers N in the category Sets. Want to linearize this setting... Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  17. 17. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions What are Natural Numbers Objects? A NNO in a CCC C consists of an object N of C together with two morphisms, zero : 1 → N and a successor mapping succ : N → N. The triple (N, zero, succ) is required to satisfy the condition that, given any pair of morphisms f : 1 → B and g : B → B in C, there exists a unique h : N → B such that the following diagram commutes. 1 zero E N d d d f d ‚ d h h c B Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva succ E N g c E B Natural Number Objects in Dialectica Categories
  18. 18. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Natural Numbers Objects in Sets A NNO in Sets consists of the object N in Sets together with two morphisms, zero : 1 → N (where we really choose the element 0 of the natural numbers N, and the successor mapping succ : N → N is really + 1 : N → N. 1 zero E N d d d f d ‚ d h h c B Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva succ E N g c E B Natural Number Objects in Dialectica Categories
  19. 19. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Natural Numbers Objects in Sets Given f : 1 −→ X (or f (∗) = x0 ) and g : X −→ X we have the map h given by h(n) = g n (f (∗)) makes the following diagram commute and is the unique map doing so. 1 zero E N +1 E N d d h d f d ‚ d c X Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva h g c E X Natural Number Objects in Dialectica Categories
  20. 20. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Natural Numbers Objects in a SMCC? Par´ and Rom´n (thinking of Linear Logic) have the following: e a Given a monoidal category C a NNO is an object N of C and morphisms zero : I −→ N and succ : N −→ N such that for any object B of C and morphisms b : I −→ B and g : B −→ B there is a morphism h : N −→ B such that the diagrams below commute: I zero E N d d d b d ‚ d succ E N h h c B g c E B Can we calculate with it? Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  21. 21. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Dialectica Categories Conceived as an internal model of G¨del’s Dialectica o Interpretation, turned out to be also a model of Linear Logic. Objects of the Dialectica category Dial2 (Sets) are triples, A = (U, X , R), where U and X are sets and R ⊆ U × X is a relation. Given elements u in U and x in X , either they are related by R, R(u, x) = 1 or they are not and R(u, x) = 0, hence the 2 in the name of the category. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  22. 22. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Dialectica Categories A morphism from A to B = (V , Y , S) is a pair of functions f : U −→ V and F : Y −→ X such that uRF (y ) =⇒ f (u)Sy . R u∈U ' X T ⇓ f c V ' Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva S F Y y Natural Number Objects in Dialectica Categories
  23. 23. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Tensor in Dialectica Categories Let A = (U, X , R) and B = (V , Y , S) be objects in Dial2 (Sets). The tensor product of A and B is given by A ⊗ B = (U × V , X V × Y U , R ⊗ S) where the relation R ⊗ S is given by (u, v ) R ⊗ S (f , g ) iff uRf (v ) and vSg (u). The unit for this tensor product is the object IDial := (1, 1, =), where 1 = {∗} is a singleton set and = is the identity relation on the singleton set. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  24. 24. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Symmetric Monoidal Closed Dialectica Categories Let A = (U, X , R) and B = (V , Y , S) be objects in Dial2 (Sets). The internal-hom is given by [A, B] = (V U × X Y , U × Y , [R, S]) where (f , F )[R, S](u, x) iff uRF (y ) implies f (u)Sy . The tensor product is adjoint to the internal-hom, as usual HomDial (A ⊗ B, C ) ∼ HomDial (A, [B, C ]) = Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  25. 25. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Other structure of Dialectica Categories Let A = (U, X , R) and B = (V , Y , S) be objects in Dial2 (Sets). There is an auxiliary tensor product structure given by A ◦ B = (U × V , X × Y , R ◦ S) where (u, v )R ◦ S(x, y ) iff uRx and vSy . This simpler tensor structure is not the adjoint of the internal-hom. It’s necessary to prove existence of appropriate modality !. The unit for this tensor product is also IDial . Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  26. 26. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Cartesian structure of Dialectica Categories Let A = (U, X , R) and B = (V , Y , S) be objects in Dial2 (Sets). The cartesian product is given by A × B = (U × V , X + Y , ch) where X + Y = X × 0 ∪ Y × 1 and the relation ch (short for ‘choose’) is given by (u, v )ch(x, 0) if uRx and (u, v )ch(y , 1) if vSy . The unit for this product is (1, ∅, ∅), the terminal object of Dial2 (Sets). (there are also cartesian coproducts.) Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  27. 27. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNOs in Dialectica Categories? To investigate iteration and recursion in linear categories we want to define a natural numbers object in Dial2 (Sets). We can use either the cartesian structure of Dial2 (Sets) or any one of its tensor structures. The first candidate monoidal structure is the cartesian product in Dial2 (Sets). This requires a map corresponding to zero from the terminal object (1, ∅, ∅) in Dial2 (Sets) to our natural numbers object candidate, say a generic object like (N, M, E ) Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  28. 28. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNOs in Dialectica Categories? Reading the definition, (N, M, E ) is a NNO with respect to the cartesian structure of Dial2 (Sets)if there are maps (z, Z ) : (1, ∅, ∅) −→ (N, M, E ) and (s, S) : (N, M, E ) −→ (N, M, E ) such that for any object (X , Y , R) and any pair of morphisms (f , F ) : (1, ∅, ∅) −→ (X , Y , R) and (g , G ) : (X , Y , R) −→ (X , Y , R) there exists some (unique) (h, H) : (N, M, E ) −→ (X , Y , R) such that a big diagram commutes. Which big diagram? Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  29. 29. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using cartesian structure ∅ ∅     1 z Z d sF d E d d d f d d ‚ d S M ' E     H T    E N s E Y G '   R c  X Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva H N h d d h M ' T Y   R c  g E X Natural Number Objects in Dialectica Categories
  30. 30. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using cartesian structure Proposition 1. The category Dial2 (Sets) has a (trivial) NNO with respect to its cartesian structure, given by (N, ∅, ∅). Proof sketch: any possible NNO for Dial2 (Sets) is of the form N = (N, M, E ) for some set M and some relation E ⊆ N × M, where N is the usual natural numbers object in Sets, with the usual zero constant and the usual successor functions. There must exist a morphism in Dial2 (Sets) zero = (z, Z ) : 1 → N with two components, z : 1 → N (as in Sets) and Z : M → 0. But since the only map into the empty set in the category of Sets is the empty map, we conclude that M is empty and so is E as this is a relation in the product N × ∅. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  31. 31. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using cartesian structure This trivial NNO works, because given any object B of Dial2 (Sets) with maps f : 1 → B and g : B → B, we can find a unique map h : N → B making all the necessary diagrams commute: In the first coordinate h is given by the map that exists for N as a NNO in Sets and in the second coordinate this is simply the empty map. Not very exciting... This triviality result is expected, since the ‘main’ structure of the category Dial2 (Sets) is the tensor product that makes it a symmetric monoidal closed category, not its cartesian structure. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  32. 32. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure [Burroni] Peano-Lawvere axiom’s says that for any object X in a zX E sX E category E , there is a diagram X NX NX with the universal property that for any diagram of the form f E g E X Y Y there exists h : NX → Y such that the following diagram commutes zX E X NX sX E NX d d h d f d ‚ d c Y Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva h g c E Y Natural Number Objects in Dialectica Categories
  33. 33. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure If a category satisfies this axiom, we say the category is a Peano-Lawvere (PL) category. Dial2 (Sets) is a PL-category if given any object (A, B, C ) of Dial2 (Sets) there is an object (N, M, E ) and maps (z, Z ) : (A, B, C ) −→ (N, M, E ) and (s, S) : (N, M, E ) −→ (N, M, E ) such that for any object of Dial2 (Sets) (X , Y , R) together with a pair of morphisms (f , F ) : (A, B, C ) −→ (X , Y , R) and (g , G ) : (X , Y , R) −→ (X , Y , R) there exists some (unique) map in Dial2 (Sets) (h, H) : (N, M, E ) −→ (X , Y , R) making the big diagram commute. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  34. 34. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure B C    A z Z d sF d E d d d f d d ‚ d S M ' E    H T    E N s E Y Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva H G '   R c  X N h d d h M ' T Y   R c  g E X Natural Number Objects in Dialectica Categories
  35. 35. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure Simplifying this picture for the case where the unit (1, 1, =) is used. If N = (N, M, E ) is a proposed NNO in Dial2 (Sets) then there must exist morphisms zero = (z, Z ) : (1, 1, =) → (N, M, E ) and succ = (s, S) : (N, M, E ) → (N, M, E ) such that: 1 =    1 z Z d sF d E d d d f d d ‚ d S M ' E    H T    E N s E Y X H G '   R c  Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva N h d d h M ' T Y   R c  X g Natural Number Objects in Dialectica Categories E
  36. 36. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure Proposition 2. The category Dial2 (Sets) has a (trivial) weak NNO with respect to its monoidal closed structure described, given by (N, 1, N × 1). As before, any possible NNO for Dial2 (Sets) is of the form N = (N, M, E ) for some set M and some relation E ⊆ N × M, where N is the usual natural numbers object in Sets, with the usual zero constant and the usual successor function on natural numbers. The morphism zero has two components, z : 1 → N (as in Sets) and Z : M → 1. The map Z has to be the unique map !M : M → 1 sending all m’s in M to the singleton set ∗ Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  37. 37. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure The morphism succ : N → N has two components (s, S), where s : N → N is the usual successor function in N, and S : M → M is to be determined, satisfying some equations. Fact 1. If there is a map (f , F ) : I → B in Dial2 (Sets) for a generic object B of the form (X , Y , R) then there exists x0 in X such for all y in Y we have x0 Ry . By definition of maps in Dial2 (Sets), we must have = 1' f c X ' 1 T F =! R Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Y Natural Number Objects in Dialectica Categories
  38. 38. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure Fact 2. If there is a NNO in Dial2 (Sets) of the form (N, M, E ), where M is the singleton set 1, then S : 1 → 1 is the identity on 1 and E relates every n in N to ∗. If N = (N, 1, E ) is a NNO in Dial2 (Sets) then the map zero = (z, Z ) : I → N has to be the zero map in N together with the terminal map in 1 and the succ = (s, S) : N → N consists of the usual successor function on the integers and S : 1 → 1 has to be the identity on 1. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  39. 39. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure The fact that (s, S) is map of Dial2 (Sets) gives us the diagram E N' 1 T S s c N' E 1 and the condition on morphisms says for all n in N and for all ∗ in 1, if nES∗ then n + 1 = s(n)E ∗. But S is the identity on 1, ie S∗ = ∗, so if nES∗ ⇒ n + 1E ∗, which is just what we need to prove that E relates every n in N to ∗. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  40. 40. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure Back to the proposition 2: The object of Dial2 (Sets) of the form (N, 1, E ), where E relates every n in N to ∗, together with morphisms zero = (0, id1 ) : I → N and succ = (+1, id1 ) : N → N is a weak NNO in Dial2 (Sets). Proof. Let B be an object (X , Y , R) of Dial2 (Sets) such that there are maps (f , F ) : I → B and (g , G ) : B → B. To prove N = (N, 1, E ), where nE ∗ for all n in N is a weak NNO, we must define a map (h, H) : N → B such that the main NNO diagram commutes. It is clear that h : N → X can be defined using the fact that N is a NNO in Sets. It is clear that we must take H : Y → 1 as the terminal map on Y . We need to check that all the required conditions are satisfied. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  41. 41. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure The 1. 2. 3. required conditions amount to showing that the map (h, H) is a map of Dial2 (Sets); the triangles commute, and the squares commute in the diagram below. 1 ' =    1 z =0 Z = id1 1 d sF =!Y E   d   E N d d d f d d ‚ d H ' T S = id1 T    E s = +1 E Y Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva H =!Y G '   R c  X N h d d h 1 Y   R c   g E X Natural Number Objects in Dialectica Categories
  42. 42. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure Items 2 and 3 are boring, but easy. Need to show that the proposed map (h, H) is indeed a map in Dial2 (Sets). The map (h, H) is a map in Dial2 (Sets) if the condition for all m in N, for all y in Y , if mEH(y ) then h(m)Ry is satisfied. Since H(y ) = ∗ and we know mE ∗ for all m in N, we need to show h(m)Ry for all m ∈ N and all y in Y . (vertical square) By cases, either m = 0 or not. Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  43. 43. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure If m = 0 we need to show h(0)Ry for all y ∈ Y . Since N is the NNO in Sets we know that 1 0 +1 E d fd ‚ E N N h h c c X g E X commutes, hence h(0) = f (∗) and h(m + 1) = g (h(m)). Since B = (X , Y , R) is an object that has a map (f , F ) : I → B we know (Fact 1) that there exists x0 = f (∗) such that f (∗)Ry for all y in Y and hence h(0) = f (∗)Ry for all y ∈ Y . Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  44. 44. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions NNO using monoidal structure If m is not zero, then m = n + 1 and h(n + 1) = g (h(n)) by the definition of h in Sets. But B is an object of Dial2 (Sets) equipped with a map (g , G ) : B → B, which means that there exist g : X → X and G : Y → Y in Sets such that for all x ∈ X and for all y ∈ Y , if xRG (y ) then g (x)Ry . To show that h(n)Ry , since we know that h(0)Ry we need to show that if h(n)Ry for all y ∈ Y then h(n + 1)Ry for all y ∈ Y . But if h(n)Ry for all y ∈ Y , then in particular h(n)RG (y ) for all y ’s that happen to be in the range of G , that is if y happens to be Gy . In this case g (h(n))Ry , that is h(n + 1)Ry . Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  45. 45. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Summing up We obtained a degenerate weak NNO, where in the first coordinate we have business as usual in Sets and in the second coordinate we have simply the singleton set 1 and terminal maps. We expected to find a NNO in the dialectica categories, with iteration and recursion as usual in the first coordinate, but co-recursion/co-iteration in the second coordinate. It is disappointing to obtain only a ‘degenerate’ NNO as above, where the second coordinate is trivial. Maybe we have not got the right level of generality... Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  46. 46. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Summing up Natural number algebras 1 → N ← N are in bijective correspondence with F -algebras (for F the endofunctor F (X ) = 1 + X ) . The initial algebra for this functor in Set is indeed the usual natural numbers, where we have an isomoprhism N ∼ 1 + N. Since this is an isomorphism we could also see it as an = F -coalgebra, but this is not final in the category of sets. As Plotkin remarks this coalgebra is final in the category of sets and partial functions Pfn. Can we change our working underlying category of Dial2 (Sets) so that a non-trivial NNO can be constructed? Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  47. 47. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Some references Sandra Alves, Maribel Fernandez, Mario Florido, and Ian Mackie. Linear recursive functions. In Rewriting, Computation and Proof, pages 182195. Springer, 2007. Dialectica and Chu constructions: Cousins?Theory and Applications of Categories, Vol. 17, 2006, No. 7, pp 127-152. Thesis TR: The Dialectica Categories http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.html A Dialectica-like Model of Linear Logic.In Proceedings of CTCS, Manchester, UK, September 1989. LNCS 389 . The Dialectica Categories. In Proc of Categories in Computer Science and Logic, Boulder, CO, 1987. Contemporary Mathematics, vol 92, American Mathematical Society, 1989 Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
  48. 48. Outline Motivation Natural Numbers Objects Dialectica Categories NNOs Conclusions Thanks! Valeria de Paiva, Charles Morgan, Samuel Gomes da Silva Natural Number Objects in Dialectica Categories
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