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ICMS 15 July disjoint work with 2020 Evan Cavallo / Carlo Angiuli Anders Mrtberg / Andrea Vezzosi Favonia floor wall floor wall wall floor comp wall wall floor cubes +) composition cubical TT major difficulty: composition

  1. ICMS 15 July disjoint work with 2020 Evan Cavallo / Carlo Angiuli Anders Mörtberg / Andrea Vezzosi Favonia

  9. floor

  10. wall floor

  11. wall wall floor

  12. comp wall wall floor

  13. cubes +) composition cubical TT major difficulty: composition for univalent universes

  17. null compositions = no walls

  18. Brunerie's number a program that should output 2* *read Guillaume Brunerie's thesis

  19. (p = (<i> <j> <k> ((test0To4 @ j) @ k) @ i))))), i = 1))))))))))))))))))))))))))))))))))))))))))) ))) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) []) 2019.03.04-cubicaltt-fbdb422ada0287dbfc7b097c4a9355ed501be6e6-stack-lts9.5-brunerie2-brunerie_opt-2.output.gz

  20. nullable compositions

  21. nullable not covering every corner not “true” under double negation not “true” under some closed substitutions

  22. kill nullable compositions!

  23. Plan A reduces to floor if null?

  24. Plan A reduces to floor if null? difficult with univalence keyword: regularity

  25. Plan B ban nullable compositions?

  26. Plan B ban nullable compositions? but universes need them in current constructions

  27. Plan C a different composition based on non-nullable ones with a different set of equations to avoid regularity

  28. Plan C a different composition based on non-nullable ones with a different set of equations to avoid regularity method 1: decision tree method 2: reflection no general construction yet

  37. cofibrations

  38. method 1: decision tree

  39. method 1: decision tree

  40. method 1: decision tree

  41. method 1: decision tree reduced reduced

  42. method 1: decision tree neocomp See [AFH] and/or Carlo's thesis

  43. method 1: decision tree neocomp comp neocomp comp comp neocomp See [AFH] and/or Carlo's thesis

  44. method 1: decision tree neocomp comp neocomp comp comp neocomp neocomp See [AFH] and/or Carlo's thesis

  45. method 1: decision tree neocomp limitation: the way/order to check dimension expressions needs to respect all equalities (e.g., subst.)

  46. method 1: decision tree variants of [AFH]-style composition D removal of duplicate walls I removal of inconsistent walls P permutation of walls S symmetry of wall constraints σ symmetry for non-diagonals only

  47. method 1: decision tree variants of [AFH]-style composition D removal of duplicate walls I removal of inconsistent walls P permutation of walls S symmetry of wall constraints σ symmetry for non-diagonals only unsolved cases: -P+S (no permutation, but with symmetry)

  48. method 1: decision tree [AFH]-style + conjunctions

  49. method 1: decision tree [AFH]-style + conjunctions trickier with +I how about

  50. method 1: decision tree [AFH]-style + conjunctions trickier with +I how about solved case by case [AFH], research notes, ...

  51. method 2: reflection [CCHM]-style composition

  52. method 2: reflection [CCHM]-style composition make intervals richer so that is surjective

  53. method 2: reflection neocomp

  54. method 2: reflection neocomp comp

  55. method 2: reflection neocomp comp used in Cubical Agda

  56. Plan C a different composition based on non-nullable ones with a different set of equations to avoid regularity but, is it worth it?

  57. none works for unknown cofibrations

  58. none works for unknown cofibrations def mycom/fun (A : 𝕁 → type) (B : 𝕁 → type) (com/A : (r : 𝕁 ) ( φ : 𝔾 ) (p : (i : 𝕁 ) (_ : [i=r ∨ φ ] (com/B : (r : 𝕁 ) ( φ : 𝔾 ) (p : (i : 𝕁 ) (_ : [i=r ∨ φ ] (r : 𝕁 ) ( φ : 𝔾 ) (p : (i : 𝕁 ) (_ : [i=r ∨ φ ]) (_ : A we can quantify over cofibrations in cooltt no known way to kill nullable compositions

  59. general theory?

  60. general theory? build univalent Kan universes with only these cofibrations still very open

  61. further reading [Angiuli] thesis Computational Semantics of Cartesian Cubical Type Theory [VMA] Cubical Agda: a dependently typed programming language with univalence and higher inductive types

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