1. Kazuyuki Asada Ichiro Hasuo RIMS, Kyoto University PRESTO Research Promotion Program, Japan Science and Technology Agency CSCAT 2010 March 18, 2010

2. Computations categorification Components

3.  Strong monad J→TK [Moggi 88]  Comonad SJ→K [Uustalu, Vene 08]  Distributive law SJ→TK  Arrow A(J,K) [Hughes 00] ↩

4. An arrow A on C [Hughes 00] is  a family of sets A(J, K) (J,K ∊ C) three families of functions  arr : Set(J, K) → A(J, K)  >>> : A(J, K)×A(K, L) → A(J, L)  first : A(J, K) → A(J×L, K×L) satisfying certain axioms, e.g. (a>>>b) >>>c = a >>>(b >>>c)

5. An arrow A on C [Hughes 00] is  a family of sets A(J, K) ∍ three families of functions  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. = a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first

6. For a (strong) monad T on C, we have a Kleisli arrow A T on C:  A T (J, K) := C T (J, K) = C(J, TK)  arr: C(J, K) → C(J, TK) (Kleisli embedding)  comp: C(J, TK)×C(K, TL) → C(J, TL) (Kleisli composition)  first: C(J, TK) → C(J×L, T(K×L)) f (f×L) ; str

7. Computations categorification Components

8. (from wikipedia)

9. C JK arr f JK c JK d KL c JK d L c JK c JK LL F: J→K : a component : Pure function : Sequential composition : Sideline c JK LL c JK : Feedback Arrow!?

10. = An arrow A on C [Hughes 00] is  a family of sets A(J, K) ∍ three families of functions  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first

11. T: a monad on Set (e.g. T = P fin finite powerset) c JK = T(X×K) J c↑ X in Set J: set of input K: set of output X: set of state Coalg ( T(-×K) J ) ∊

12. c JK d KL c JK d L >>> ＝, ＝＝＝＝ T(X×K) J a↑ X T(Y×L) K b↑ Y T((X×Y)×L) J ↑ X×Y a>>>b state spaces: (X×Y) ×Z ≅ X× (Y×Z) ≅ c JK d L e M d KL e M c J

13. An arrow A on C is  a family of sets A(J, K) ∍ three families of functions  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first =

14. ≅ An categorical arrow A on C is  a family of sets A(J, K) ∍ three families of functions  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first

15. ≅ An categorical arrow A on C is  a family of categories A(J, K) ∍ three families of functions  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first Coalg ( T(-×K) J )

16. ≅ An categorical arrow A on C is  a family of categories A(J, K) ∍ three families of functors  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first Coalg ( T(-×K) J )

17. ≅ An categorical arrow A on C is  a family of categories A(J, K) ∍ three families of functors  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first

18.  When A(J, K) ( = { c } ) := Coalg ( T(-×K) J ) JK T(X×K) J c↑ X T(Z×K) J ↑final Z - - - - ＞ ＞ behavior

19. Computations categorification ArrowCategorical Arrow Components

20. Computations categorification ArrowCategorical Arrow Components

21. Computations categorification ArrowCategorical Arrow Components

22. Computations categorification ArrowCategorical Arrows Components Arrows concrete construction

23. Main technical result:  For given arrow A on Set,  we construct an arrow-based coalgebraic model: ∊ Coalg ( A(J, -×K) ),  we show that these Coalg ( A(J, -×K) ) (J,K ∊ Set) form a categorical arrow. A(J, X×K) c↑ X ≅ c JK d L e M d KL e M c J

24.  arrow A categ. arrow Coalg ( A(J, -×K) ) monad T categ. arrow Coalg ( T(-×K) J ) [Barbosa]+[Hasuo, H., J., S. 09]  a proof technique of calculation in Prof (the bicategory of profunctors), using the fact: Arrow is strong monad in Prof. () generalization

25. What is Prof ?

26.  A profunctor F: C ―|―> D (C, D: categories) is a functor F: D op ×C → Set Y, X F(Y, X).

27.  A profunctor F: C ――> D (C, D: categories) is a functor F: D op ×C → Set Y, X F(Y, X). p =  For g: Y’ → Y in D, f: X → X’ in C, F(g, f) (p) = p YX ∊ Y p X g f Y’X’

28.  Composition G 。 F: C ――> E is E op ×C → Set Z, X Σ Y { (, ) } / ~ (sets of “formal composition”) p YX p YX q ZY A profunctor F: C ――> D is a functor F: D op ×C → Set Y, X F(Y, X) = { }

29. σ Prof is a bicategory whose  0-cells are small categories,  1-cells are profunctors, and  2-cells are natural transformations. ( C D ：＝ D op ×C Set ) ⇒ σ ⇒

30. Why Prof ?

31. F α X Cα ⇓ D vs. FX GX G

32. MonadArrow(w/o first) T: C → C η X : X→TX μ X : TTX→TX TTT ――> TT ↓ ↓ TT ――> T … equipped with satisfying A:C op ×C→Set arr: C(J,K)→A(J,K) >>>: A(J,K) × A(K,L) → A(J,L) (a >>> b) >>> c = a >>> (b >>> c) …

33. MonadArrow(w/o first) T: C C C C C C C C C C C = C C C C … equipped with satisfying A:C op ×C→Set arr: C(J,K)→A(J,K) >>>: A(J,K) × A(K,L) → A(J,L) (a >>> b) >>> c = a >>> (b >>> c) … η ⇒ μ ⇒ μ ⇒ μ ⇒ μ ⇒ μ ⇒ in Cat

34. MonadArrow(w/o first) T: C C C C C C C C C C C = C C C C … equipped with satisfying A: C C C C C C C C C C C = C C C C … η ⇒ μ ⇒ μ ⇒ μ ⇒ μ ⇒ μ ⇒ arr ⇒ ⇒ ⇒ ⇒ ⇒⇒ >>> [Jacobs, H., H. 09] in Catin Prof

35. MonadArrow equipped with satisfying first: A(J,K) → A(J×L, K×L ) …

36. MonadArrow C×C C×C C C … equipped with Satisfying C×C C×C C C … A×C ×× A ⇒ first T×C ×× T ⇒ str [Asada 10] in Catin Prof

37.  Arrow is equivalent to strong monad in Prof: i.e., C C satisfying familiar axioms. A×C ×× ⇒ first A C×C C×C C C arr ⇒ C ⇒ >>> C A A A AA in Prof

38. Theorem For an arrow A, the categories Coalg ( A(J, -×K) ) (J,K ∊ Set) forms a categorical arrow.

39. ≅ An categorical arrow A on C is  a family of categories A (J, K) ∍ three families of functors  (Embedding a pure function f:J →K)  (Sequential composition),  (Sideline) satisfying certain axioms, e.g. a JK arr f JK a JK b KL a JK b L a JK a JK LL >>> a JK b L c M b KL c M a J first Coalg ( A(J, -×K) )

40. Proof. Lemma For an arrow A, the endofunctors A ( J, -×K ) (J,K ∊ Set) forms a lax arrow functor. Lemma [Hasuo, H., J., S. 09] If (F J,K ) J,K is a lax arrow endofunctor, Then Coalg ( F J,K ) (J,K ∊ Set) forms a categorical arrow.

41. Proof. Lemma For an arrow A, the endofunctors A ( J, -×K ) (J,K ∊ Set) forms a lax arrow functor. Lem [Hasuo, J., H., S. 09] If (F J,K ) J,K is a lax arrow endofunctor, Then Coalg ( F J,K ) (J,K ∊ Set) forms a categorical arrow.

42.  (F J,K : Set→Set) J,K is a lax arrow functor if it is equipped with  F arr f : 1 → F J,K (1)  F >>> J,K,L : F J,K (X)×F K,L (Y) → F J,L (X×Y)  F first J,K,L : F J,K (X) → F J × L, K × L (X) satisfying certain axioms (similar to arrow’s). cf. F 1 : 1 → F1 F × : FX×FY → F(X×Y)

43. LemmaFor an arrow A, there are the following structures:  F arr f : 1 → A(J, 1×K)  F >>> J,K,L : A(J, X×K)× A(K, Y×L) → A(J, (X×Y)×L)  F first J,K,L : F J,K (X) → F J × L, K × L (X) satisfying certain axioms (similar to arrow’s).

44. E.g. A(J, X×K)× A(K, Y×L) → A(J, (X×Y)×L) can be described in Prof as F >>> J,K,L C3C2C2C2CCCC3C2C2C2CCC := C3C2C2C2CCCC3C2C2C2CCC × A F >>> ⇒⇒⇒ ⇒ >>> ≅ second C×A A A C× (×) (×)× C × × × C×A A C× (×) (×)× C ×

45. C3C2C2C2CCCC3C2C2C2CCC := C3C2C2C2CCCC3C2C2C2CCC × A F >>> ⇒⇒⇒ ⇒ >>> ≅ second C×A A A C× (×) (×)× C × × × C×A A C× (×) (×)× C × C4C3C3C3C3C2C2C2C2CCCC4C3C3C3C3C2C2C2C2CCC ⇒ ≅ C4C3C3C2C2C3C2C2CCCC2CC4C3C3C2C2C3C2C2CCCC2C F >>> ⇒ ⇒ C× F >>> ⇒ F >>> ⇒ C× (×) C 2 × (×) (×)× C 2 C×AC 2 ×AC× (×) (×)× C × C×A × A × (×)× C A A (×)× C 2 (×)× C × A A × C×A C× (×) C 2 ×AC 2 × (×) C×A C× (×) (×)× C C ×(×)× C ＝

46. Computations categorification ArrowCategorical Arrow(s) Components Arrows concrete construction

47. Key lemma Microcosm principles Main theorem [Hasuo, H., J., S. 09]

48. Key lemma Microcosm principles Outer model (main theorem) Inner model compositionality [Hasuo, H., J., S. 09]