a sequent calculus for opetopes
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A sequent calculus for opetopes CT 2019 Pierre-Louis Curien 1 Cdric Ho Thanh 1 Samuel Mimram 2 July 9th, 2019 1 IRIF, Paris University 2 LIX, cole Polytechnique This presentation informally presents some of the main notions and results of

  1. Idea: dimension 0 and 1 Denote by ⧫ the unique 0-opetope, a.k.a. the point: . and by ◾ the unique 1-opetope, a.k.a. the arrow: . . We can represent ◾ as a node of a tree as follows: ⧫ ∗ [] ◾ ⧫ Let us add address information. 11

  2. 2. consider that tree as a corolla, where the input edges are the nodes 3. be convinced that this is a good representation of some 2-opetope! Idea: dimension 2 Then we can: 1. create a tree with that corolla representing ◾ ⧫ ∗ [ ∗∗ ] ◾ ⧫ ∗ [ ∗ ] ◾ ⧫ ∗ [] ◾ ⧫ 12

  3. 3. be convinced that this is a good representation of some 2-opetope! Idea: dimension 2 Then we can: 1. create a tree with that corolla representing ◾ ⧫ ∗ [ ∗∗ ] ◾ [ ] ∗ [ ∗ ] [ ∗ ] ⧫ ∗ [ ∗ ] ◾ ◾ ◾ [] ◾ 3 ⧫ ∗ [] ◾ ◾ ⧫ 2. consider that tree as a corolla, where the input edges are the nodes 12

  4. Idea: dimension 2 Then we can: 1. create a tree with that corolla representing ◾ ⧫ ∗ [ ∗∗ ] ◾ [ ] ∗ [ ∗ ] . . [ ∗ ] ⧫ ∗ [ ∗ ] ◾ ◾ ◾ [] ⇓ ◾ 3 . . ⧫ ∗ [] ◾ ◾ ⧫ 2. consider that tree as a corolla, where the input edges are the nodes 3. be convinced that this is a good representation of some 2-opetope! 12

  5. Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ⧫ ∗ [ [ ∗ ] ] ∗ [ ] . ◾ ◾ ◾ [] ⟿ ⟿ ⧫ ∗ ⇓ [] 2 . . ◾ ◾ ⧫ 13

  6. Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ⧫ ∗ [ ∗∗⋯∗ ] ◾ ⧫ ∗ [ ∗ ] ∗ [ ∗ ] ⋯ [ ⋯ ∗ ] ⋮ ( n − 1 ) . . . ◾ ◾ ◾ [] ⟿ ⟿ n ( n ) ⇓ ( 1 ) . . ⧫ ∗ [ ∗ ] ◾ ◾ ⧫ ∗ [] [] ◾ ⧫ 13

  7. Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: ◾ ⧫ [] ∗ [] [] 1 ⟿ ⟿ ◾ ⇓ . . ⧫ ◾ 13

  8. Idea: dimension 2 Depending on the original tree, we obtain different 2-opetopes: [] . 0 ⟿ ⟿ ⧫ ⇓ ◾ 13

  9. Idea: dimension 3 From there, repeat the process! [ ] ] ∗ ◾ ◾ [ [ [[ ∗ ]] ] ] ] 2 2 2 ∗ [ [ [ ] [] ⟿ A ◾ ◾ [ ∗ ] [] 2 3 ◾ . . . . ⟿ ⇓ ⇓ ⇛ ⇓ . . . . 14

  10. Idea: dimension 3 From there, repeat the process! [[]] 0 [ ] ] 0 1 ] [ ◾ [] [ [] [] ⟿ B 1 0 ◾ . . ⇓ ⟿ ⇛ ⇓ ⇓ 14

  11. Idea: dimension 3 From there, repeat the process! [[ ∗ ]] 0 [ [ ] ] ] 0 ] 2 ∗ ◾ ◾ [ [ [ ∗ ] [] [] ⟿ C 2 0 ◾ . . . . ⇓ ⟿ ⇓ ⇓ ⇛ 14

  12. Idea: dimension 3 From there, repeat the process! [ ] ] ◾ ◾ ∗ ◾ [ [] [ ] [[ ∗ ]] [[ ∗∗ ]] [[ ∗ ]] ] 2 1 3 2 1 ] ∗ [ ] [ ∗ [ [] ⟿ D ◾ ◾ ◾ [ ∗ ] [] [ ∗∗ ] 3 4 ◾ . . ⇓ ⟿ ⇛ . . . . ⇓ ⇓ ⇓ . . . . 14

  13. Solution In an n -opetope, every node is decorated by n 1 -opetope, but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactic description of such trees. 15

  14. but n 1 -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactic description of such trees. Solution ◾ ◾ 2 . . . . ⟿ ◾ ◾ ⇓ ⇓ ⇛ ⇓ 2 . . . . ◾ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, 15

  15. But addresses do! So we just need to describe a partial map n 1 Syntax We now want a syntactic description of such trees. Solution ◾ ◾ 2 . . . . ⟿ ◾ ◾ ⇓ ⇓ ⇛ ⇓ 2 . . . . ◾ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, but ( n − 1 ) -opetope does not uniquely identify a node. 15

  16. Syntax We now want a syntactic description of such trees. Solution [ ] ] ◾ ◾ ∗ [ [[ ∗ ]] 2 [ ] . . . . ⟿ ⇓ ◾ ⇓ ◾ ⇛ [ ∗ ] [] ⇓ 2 . . . . ◾ In an n -opetope, every node is decorated by ( n − 1 ) -opetope, but ( n − 1 ) -opetope does not uniquely identify a node. But addresses do! So we just need to describe a partial map 15 A � → O n − 1 .

  17. Syntax We encode opetopes recursively as follows: [ ] ] ∗ ⎧ ◾ ◾ [ ⎪ [[ ∗ ]] [] ← 2 ⎪ 2 ⎨ [ ] ⎪ [[ ∗ ]] ← 2 ⎪ ⟿ ⎩ ◾ ◾ [ ∗ ] [] 2 ◾ 16

  18. Syntax We encode opetopes recursively as follows: [ ] ] ∗ ⎧ ◾ ◾ [ ⎪ [[ ∗ ]] [] ← 2 ⎪ 2 ⎨ [ ] ⎪ [[ ∗ ]] ← 2 ⎪ ⟿ ⎩ ◾ ◾ [ ∗ ] [] 2 ◾ Reminder ⧫ ∗ [ ∗ ] ◾ 2 = ⧫ ∗ [] ◾ 16 ⧫

  19. Syntax We encode opetopes recursively as follows: [ ] ] ∗ ⎧ ◾ ◾ [ ⎪ [[ ∗ ]] [] ← 2 ⎪ 2 ⎨ [ ] ⎪ [[ ∗ ]] ← 2 ⎪ ⟿ ⎩ ◾ ◾ [ ∗ ] [] 2 ◾ Reminder ⧫ ⎧ ∗ [ ∗ ] ⎪ ⎪ [] ← ◾ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ 16 ⧫

  20. Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [ [] ← ◾ ⎪ ⎪ ] ⎪ ⎪ ] [] ← ⎨ ⎪ ∗ ◾ ◾ ⎪ [ ⎪ ⎪ [ ∗ ] ← ◾ [[ ∗ ]] ⎪ ⎪ ⎩ 2 ⎨ ⎧ [ ] ⎪ ⎪ ⎪ ⎪ [] ← ◾ ⟿ ⎪ ⎪ ◾ ◾ [[ ∗ ]] ← ⎨ ⎪ [ ∗ ] [] ⎪ ⎪ ⎪ [ ∗ ] ← ◾ ⎪ ⎪ 2 ⎩ ⎩ ◾ Reminder ⧫ ⎧ ∗ [ ∗ ] ⎪ ⎪ [] ← ◾ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ 16 ⧫

  21. Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [ [] ← ◾ ⎪ ⎪ ] ⎪ ⎪ ] [] ← ⎨ ⎪ ∗ ◾ ◾ ⎪ [ ⎪ ⎪ [ ∗ ] ← ◾ [[ ∗ ]] ⎪ ⎪ ⎩ 2 ⎨ ⎧ [ ] ⎪ ⎪ ⎪ ⎪ [] ← ◾ ⟿ ⎪ ⎪ ◾ ◾ [[ ∗ ]] ← ⎨ ⎪ [ ∗ ] [] ⎪ ⎪ ⎪ [ ∗ ] ← ◾ ⎪ ⎪ 2 ⎩ ⎩ ◾ Convention { ∗ ← ⧫ ◾ = 16

  22. Syntax We encode opetopes recursively as follows: ⎧ ⎧ ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ [ ⎪ ⎪ ] ⎪ ⎪ [] ← ⎨ ] ⎪ ∗ ◾ ◾ ⎪ [ ⎪ ⎪ [ ∗ ] ← { ∗ ← ⧫ ⎪ [[ ∗ ]] ⎪ ⎩ 2 ⎨ [ ⎧ ] ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⟿ ⎪ ⎪ ⎪ ◾ ◾ ⎪ [[ ∗ ]] ← ⎨ [ ∗ ] [] ⎪ ⎪ ⎪ ⎪ [ ∗ ] ← { ∗ ← ⧫ 2 ⎪ ⎪ ⎩ ⎩ ◾ Convention { ∗ ← ⧫ ◾ = 16

  23. 1 0 Syntax: examples [[]] 0 ◾ [] [] 1 ◾ 17

  24. Syntax: examples [[]] 0 ⎧ ⎪ ⎪ [] ← 1 ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ 17

  25. Syntax: examples [[]] 0 ⎧ ⎪ ⎪ [] ← 1 ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ Reminder ⧫ {[] ← ◾ ∗ [] 1 = = ◾ ⧫ 17

  26. Syntax: examples [[]] 0 ⎧ ⎪ [] ← {[] ← ◾ ⎪ ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ Reminder ⧫ {[] ← ◾ ∗ [] 1 = = ◾ ⧫ 17

  27. Syntax: examples [[]] 0 ⎧ ⎪ [] ← {[] ← ◾ ⎪ ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ Reminder { ∗ ← ⧫ ◾ = 17

  28. Syntax: examples [[]] 0 ⎧ ⎪ [] ← {[] ← { ∗ ← ⧫ ⎪ ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ Reminder { ∗ ← ⧫ ◾ = 17

  29. Syntax: examples [[]] 0 ⎧ ⎪ [] ← {[] ← { ∗ ← ⧫ ⎪ ⎨ ◾ [] [] ⎪ [[]] ← 0 ⎪ ⟿ ⎩ 1 ◾ Reminder + convention { { ⧫ 0 = = ⧫ 17

  30. Syntax: examples [[]] 0 ⎧ ⎪ [] ← {[] ← { ∗ ← ⧫ ⎪ ⎨ ◾ [] [] ⎪ [[]] ← { { ⧫ ⎪ ⟿ ⎩ 1 ◾ Reminder + convention { { ⧫ 0 = = ⧫ 17

  31. 2 0 Syntax: examples [[ ∗ ]] 0 [ ] ◾ ◾ [ ∗ ] [] 2 ◾ 18

  32. Syntax: examples [[ ∗ ]] 0 ⎧ [ ⎪ ] ⎪ [] ← 2 ⎨ ◾ ◾ [ ∗ ] [] ⎪ [[ ∗ ]] ← 0 ⎪ ⟿ ⎩ 2 ◾ 18

  33. Syntax: examples [[ ∗ ]] 0 ⎧ [ ⎪ ] ⎪ [] ← 2 ⎨ ◾ ◾ [ ∗ ] [] ⎪ [[ ∗ ]] ← 0 ⎪ ⟿ ⎩ 2 ◾ Reminder ⧫ ⎧ ∗ [ ∗ ] ⎪ ⎪ [] ← ◾ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ ⧫ 18

  34. Syntax: examples ⎧ [[ ∗ ]] ⎧ ⎪ ⎪ ⎪ [] ← ◾ 0 ⎪ [ ⎪ [] ← ⎨ ] ⎪ ⎪ ⎨ [ ∗ ] ← ◾ ◾ ⎪ ◾ [ ∗ ] ⎩ [] ⎪ ⎪ ⟿ ⎪ 2 ⎪ [[ ∗ ]] ← 0 ⎩ ◾ Reminder ⧫ ⎧ ∗ [ ∗ ] ⎪ ⎪ [] ← ◾ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ ⧫ 18

  35. Syntax: examples ⎧ [[ ∗ ]] ⎧ ⎪ ⎪ ⎪ [] ← ◾ 0 ⎪ [ ⎪ [] ← ⎨ ] ⎪ ⎪ ⎨ [ ∗ ] ← ◾ ◾ ⎪ ◾ [ ∗ ] ⎩ [] ⎪ ⎪ ⟿ ⎪ 2 ⎪ [[ ∗ ]] ← 0 ⎩ ◾ Reminder { ∗ ← ⧫ ◾ = 18

  36. Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ 0 ⎪ [ ⎪ [] ← ⎨ ] ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ◾ ⎪ ◾ [ ∗ ] ⎩ [] ⎪ ⎪ ⟿ ⎪ 2 ⎪ ⎪ [[ ∗ ]] ← 0 ⎩ ◾ Reminder { ∗ ← ⧫ ◾ = 18

  37. Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ 0 ⎪ [ ⎪ [] ← ⎨ ] ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ◾ ⎪ ◾ [ ∗ ] ⎩ [] ⎪ ⎪ ⟿ ⎪ 2 ⎪ ⎪ [[ ∗ ]] ← 0 ⎩ ◾ Reminder { { ⧫ 0 = = ⧫ 18

  38. Syntax: examples ⎧ ⎧ [[ ∗ ]] ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ 0 ⎪ [ ⎪ [] ← ⎨ ] ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ ◾ ◾ ⎩ [ ∗ ] [] ⎪ ⎪ ⟿ ⎪ 2 ⎪ ⎪ [[ ∗ ]] ← { { ⧫ ⎩ ◾ Reminder { { ⧫ 0 = = ⧫ 18

  39. 3 2 1 Syntax: examples [ ] ] ∗ ◾ ◾ ◾ [ [] [[ ∗ ]] [[ ∗∗ ]] 2 1 [ ] ◾ ◾ ◾ [ ∗ ] [] [ ∗∗ ] 3 ◾ 19

  40. Syntax: examples [ ] ⎧ ] ⎪ [] ← 3 ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ 2 1 ⎨ [[ ∗ ]] ← 2 [ ] ⎪ ⎪ ⟿ ⎪ ⎪ ◾ ◾ ◾ [[ ∗∗ ]] ← 1 [ ∗ ] ⎩ [] [ ∗∗ ] 3 ◾ 19

  41. Syntax: examples [ ] ⎧ ] ⎪ [] ← 3 ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ 2 1 ⎨ [[ ∗ ]] ← 2 [ ] ⎪ ⎪ ⟿ ⎪ ⎪ ◾ ◾ ◾ [[ ∗∗ ]] ← 1 [ ∗ ] ⎩ [] [ ∗∗ ] 3 ◾ Reminder ⧫ ∗ [ ∗∗ ] ⎧ [] ← ◾ ⎪ ⎪ ◾ ⎪ ⎪ ⎨ [ ∗ ] ← ◾ ⧫ [ ∗ ] ∗ 3 ⎪ ⎪ = = ◾ ⎪ ⎪ [ ∗∗ ] ← ◾ ⎩ ⧫ [] ∗ ◾ ⧫ 19

  42. Syntax: examples ⎧ ⎧ ⎪ ⎪ [] ← ◾ ⎪ ⎪ ⎪ ⎪ [ ⎪ ] ⎪ ⎪ ⎪ ] ⎪ [] ← ⎨ [ ∗ ] ← ◾ ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎪ ⎪ 2 1 ⎨ ⎪ [ ∗∗ ] ← ◾ [ ⎩ ] ⎪ ⎪ ⟿ ⎪ ⎪ ◾ ◾ ◾ [[ ∗ ]] ← 2 [ ∗ ] ⎪ [] [ ∗∗ ] ⎪ ⎪ ⎪ 3 ⎪ ⎪ [[ ∗∗ ]] ← 1 ⎩ ◾ Reminder ⧫ [ ∗∗ ] ∗ ⎧ ⎪ [] ← ◾ ⎪ ◾ ⎪ ⎪ ⎨ [ ∗ ] ← ◾ ⧫ [ ∗ ] ∗ 3 ⎪ ⎪ = = ⎪ ◾ ⎪ [ ∗∗ ] ← ◾ ⎩ ⧫ [] ∗ ◾ ⧫ 19

  43. Syntax: examples ⎧ ⎧ ⎪ ⎪ [] ← ◾ ⎪ ⎪ ⎪ ⎪ [ ⎪ ] ⎪ ⎪ ⎪ ] ⎪ [] ← ⎨ [ ∗ ] ← ◾ ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎪ ⎪ 2 1 ⎨ ⎪ [ ∗∗ ] ← ◾ [ ⎩ ] ⎪ ⎪ ⟿ ⎪ ⎪ ◾ ◾ ◾ [[ ∗ ]] ← 2 [ ∗ ] ⎪ [] [ ∗∗ ] ⎪ ⎪ ⎪ 3 ⎪ ⎪ [[ ∗∗ ]] ← 1 ⎩ ◾ Reminder ⧫ {[] ← ◾ ∗ [] 1 = = ◾ ⧫ 19

  44. Syntax: examples ⎧ ⎧ [] ← ◾ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ ⎪ ⎪ ] ⎪ ⎪ ] [] ← ⎨ [ ∗ ] ← ◾ ⎪ ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎪ ⎪ ⎪ 2 1 ⎨ [ ∗∗ ] ← ◾ [ ⎩ ] ⎪ ⎪ ⟿ ⎪ ⎪ [[ ∗ ]] ← 2 ◾ ◾ ◾ [ ∗ ] ⎪ [] [ ∗∗ ] ⎪ ⎪ ⎪ 3 ⎪ ⎪ [[ ∗∗ ]] ← {[] ← ◾ ⎩ ◾ Reminder ⧫ {[] ← ◾ ∗ [] 1 = = ◾ ⧫ 19

  45. Syntax: examples ⎧ ⎧ [] ← ◾ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ ⎪ ⎪ ] ⎪ ⎪ ] [] ← ⎨ [ ∗ ] ← ◾ ⎪ ∗ ◾ ◾ ◾ ⎪ [ [] ⎪ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎪ ⎪ ⎪ 2 1 ⎨ [ ∗∗ ] ← ◾ [ ⎩ ] ⎪ ⎪ ⟿ ⎪ ⎪ [[ ∗ ]] ← 2 ◾ ◾ ◾ [ ∗ ] ⎪ [] [ ∗∗ ] ⎪ ⎪ ⎪ 3 ⎪ ⎪ [[ ∗∗ ]] ← {[] ← ◾ ⎩ ◾ Reminder ⧫ ⎧ ∗ [ ∗ ] ⎪ ⎪ [] ← ◾ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ ⧫ 19

  46. Syntax: examples ⎧ ⎧ ⎪ [] ← ◾ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [] ← ⎨ [ ∗ ] ← ◾ [ ⎪ ] ⎪ ⎪ ⎪ ] ⎪ ⎪ ◾ ◾ ∗ ◾ ⎪ [] ⎪ [ ⎪ [ ∗∗ ] ← ◾ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎩ 1 2 ⎨ ⎧ [ ] ⎪ ⎪ [] ← ◾ ⎪ ⎪ ⟿ ⎪ [[ ∗ ]] ← ⎨ ⎪ ◾ ◾ ◾ [ ∗ ] ⎪ [] ⎪ [ ∗∗ ] ⎪ ⎪ [ ∗ ] ← ◾ ⎪ ⎪ 3 ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ [[ ∗∗ ]] ← {[] ← ◾ ⎩ ◾ Reminder ⧫ ∗ ⎧ [ ∗ ] ⎪ [] ← ◾ ⎪ ◾ ⎨ 2 ⎪ [ ∗ ] ← ◾ ⎪ = = ⧫ ⎩ ∗ [] ◾ ⧫ 19

  47. Syntax: examples ⎧ ⎧ ⎪ [] ← ◾ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [] ← ⎨ [ ∗ ] ← ◾ [ ⎪ ] ⎪ ⎪ ⎪ ] ⎪ ⎪ ◾ ◾ ∗ ◾ ⎪ [] ⎪ [ ⎪ [ ∗∗ ] ← ◾ ⎪ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎩ 1 2 ⎨ ⎧ [ ] ⎪ ⎪ [] ← ◾ ⎪ ⎪ ⟿ ⎪ [[ ∗ ]] ← ⎨ ⎪ ◾ ◾ ◾ [ ∗ ] ⎪ [] ⎪ [ ∗∗ ] ⎪ ⎪ [ ∗ ] ← ◾ ⎪ ⎪ 3 ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ [[ ∗∗ ]] ← {[] ← ◾ ⎩ ◾ Reminder { ∗ ← ⧫ ◾ = 19

  48. Syntax: examples ⎧ ⎧ ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [] ← ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ [ ⎪ ] ⎪ ⎪ ⎪ ⎪ ] ⎪ ⎪ ◾ ◾ ∗ ◾ [] ⎪ ⎪ [ ⎪ ⎪ [ ∗∗ ] ← { ∗ ← ⧫ [[ ∗ ]] [[ ∗∗ ]] ⎪ ⎩ 2 1 ⎨ [ ] ⎧ ⎪ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ ⟿ ⎪ ⎪ ◾ ◾ ◾ [ ∗ ] ⎪ [[ ∗ ]] ← ⎨ [] ⎪ [ ∗∗ ] ⎪ ⎪ ⎪ [ ∗ ] ← { ∗ ← ⧫ 3 ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ◾ ⎪ [[ ∗∗ ]] ← {[] ← { ∗ ← ⧫ ⎩ Reminder { ∗ ← ⧫ ◾ = 19

  49. Syntax Question Is this an opetope? ⎧ ⎧ [ ∗ ] ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [] ← ⎨ [ ∗∗ ] ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ ∗ ∗ ∗ ] ← ⧫ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ [ ∗∗ ] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← {[] ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎧ ⎪ ⎪ [] ← {[] ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [] ← ⎨ ⎪ [ ∗ ] ← ⧫ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ ∗∗ ] ← ⧫ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ [ ∗ ∗ ∗ ] ← ⎨ ⎪ ⎪ ⎪ ⎪ [[]] ← {[] ← { ∗ ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ [[[ ∗ ]]] ← { ∗ ← ⧫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [[[]]] ← ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [ ∗ ] ← { ∗ ← ⧫ ⎪ ⎩ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ [[ ∗ ∗ ∗ ]] ← ⧫ ⎩ 20

  50. Opt ? : a sequent calculus for opetopes

  51. System Opt aims to characterize preopetopes that actually are opetopes. System Opt ? The set of preopetopes P is defined by the following grammar: ::= P ⧫ ⎧ ⎪ ⎪ ⎪ ⎪ A ← P ⎨ | ⎪ ⎪ ⋮ ⎪ ⎪ ⎩ A ← P { { P | 21

  52. System Opt ? The set of preopetopes P is defined by the following grammar: ::= P ⧫ ⎧ ⎪ ⎪ ⎪ ⎪ A ← P ⎨ | ⎪ ⎪ ⋮ ⎪ ⎪ ⎩ A ← P { { P | System Opt ? aims to characterize preopetopes that actually are opetopes. 21

  53. System Opt ? : the point rule The first rule of Opt ? states that we may create points without any prior assumption: point point ⧫ . 22

  54. System Opt ? : the shift rule This rule takes an opetope p and produces a new opetope having a unique node, decorated in p : [ ] ] ◾ ◾ ∗ [ [[ ∗ ]] 2 [ ] ◾ ◾ p [ ∗ ] [] 2 shift {[] ← p ◾ shift [ ] ] ] 2 2 ∗ [ [ [] A 3 23

  55. System Opt ? : the degen rule This rule takes an opetope and produces a degenerate opetope from it: p . degen degen . { { p ⇓ 24

  56. System Opt ? : the graft rule This rule glues an n -opetope q to an ( n + 1 ) -opetope p , the latter really just being a pasting diagram of n -opetopes, and “glues” them together: ⎧ [ a 1 ] ← r 1 ⎪ ⎪ ⎪ ⎪ ⎨ . q ⎪ ⎪ ⋮ ⇓ ⎪ ⇓ ⎪ [ a k ] ← r k ⎩ . . . . ⇓ graft- [ b ] ⎧ ⎪ [ a 1 ] ← r 1 . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⇓ ⎨ ⋮ . . . ⎪ ⇓ [ a k ] ← r k ⎪ ⇓ ⎪ ⎪ ⎪ ⎪ . . ⎪ [ b ] ← q ⎩ 25

  57. Main result Theorem Derivable preopetopes in system Opt ? are in bijective correspondence with opetopes. 26

  58. Examples

  59. Examples The proof tree of ⧫ = . is point ⧫ 27

  60. Examples The proof tree of ◾ = . . is point ⧫ shift {[] ← ⧫ 27

  61. Examples The proof tree of ◾ = . . is point ⧫ shift { ∗ ← ⧫ 27

  62. Examples The proof tree of ⧫ ∗ [] 1 = ⟿ ◾ ⇓ . . ⧫ is point ⧫ shift { ∗ ← ⧫ shift {[] ← { ∗ ← ⧫ 27

  63. Examples The proof tree of ⧫ ∗ [ ∗ ] . ◾ 2 = ⟿ ⧫ ∗ ⇓ [] . . ◾ ⧫ is point ⧫ shift { ∗ ← ⧫ point ⧫ shift shift {[] ← { ∗ ← ⧫ {[] ← ⧫ graft- [ ∗ ] {[] ← { ∗ ← ⧫ [ ∗ ] ← { ∗ ← ⧫ 27

  64. Examples The proof tree of [ ∗∗ ] ⧫ ∗ . . ◾ [ ∗ ] ⧫ 3 ∗ = ⇓ ⟿ ◾ [] . . ⧫ ∗ ◾ is ⧫ point ⧫ shift { ∗ ← ⧫ point ⧫ shift shift {[] ← { ∗ ← ⧫ {[] ← ⧫ graft- [ ∗ ] point {[] ← { ∗ ← ⧫ ⧫ shift {[] ← ⧫ [ ∗ ] ← { ∗ ← ⧫ graft- [ ∗∗ ] ⎧ ⎪ [] ← { ∗ ← ⧫ ⎪ ⎪ ⎪ ⎨ [ ∗ ] ← { ∗ ← ⧫ ⎪ ⎪ ⎪ ⎪ [ ∗∗ ] ← { ∗ ← ⧫ ⎩ 27

  65. Examples The proof tree of [ ] ] ◾ ◾ ∗ [ [[ ∗ ]] 2 [ ] . . . . ⇓ ⟿ ⇓ ◾ ◾ ⇛ [ ∗ ] [] ⇓ 2 . . . . ◾ is: ⋮ 2 28

  66. Examples The proof tree of [ ] ] ◾ ◾ ∗ [ [[ ∗ ]] 2 [ ] . . . . ⇓ ⟿ ⇓ ◾ ◾ ⇛ [ ∗ ] [] ⇓ 2 . . . . ◾ is: ⋮ 2 shift {[] ← 2 28

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