The Internal n-Cubes #902
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The Internal n-Cubes #902
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ecavallo
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Aug 29, 2022
Cubical/Foundations/Cubes/Base.agda
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| -- A cube is just a path of cubes of one-lower-dimension. | ||
| -- Unfortunately the following function cannot begin at 0, | ||
| -- because Agda doesn't support pattern matching on ℕ towards pre-types. | ||
| pathCube : (n : ℕ) → (I → Cube (suc n) A) → Cube (suc (suc n)) A | ||
| pathCube n p = _ , λ i → p i .snd |
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Let's add a reminder here that we can fix this in 2.6.3 when I -> A becomes fibrant.
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| setConst : {n : ℕ}{A : Type ℓ} → (cube : Cube (suc n) A) → constCube (∂₀ cube) ≡ cube | ||
| setConst {n = 0} (_ , p) i = _ , λ j → p (i ∧ j) | ||
| setConst {n = suc n} (_ , p) i = _ , λ j → p (i ∧ j) |
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Is set short for section?
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Looks good to me! |
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Ehhh.... How could I write 2.9.3? |
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This PR contains a mutual recursive definition of n-cubes, their boundaries and n-cubes with fixed boundary, as family of types parametrized over
ℕ. They're strictly isomorphic to external ones as, for example,(i j : I) → A,(i j : I) → Partial (i ∨ ~ i ∨ j ∨ ~ j) Aand(i j : I) → A [ (i ∨ ~ i ∨ j ∨ ~ j) ↦ u i j ]. Unless one fixes a canonical natural numbernas dimension, these isomorphisms cannot be described internally, so for more general results one must resort to macros or 2LTT.My aim is to prove that a type has h-level
nif and only if n-boundaries can always be filled into n-cubes, generalizing the special cases aboutisSet'andisGroupoid'already existing in library.