Trying to prove cummulativity of homotopy levels

This commit is contained in:
Frederik Hanghøj Iversen 2018-03-12 16:00:27 +01:00
parent c52384b012
commit fe453a6d3a
2 changed files with 41 additions and 9 deletions

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@ -175,12 +175,12 @@ module voe {a b : Level} ( : Category a b) where
where where
lem : Monoidal→Kleisli Kleisli→Monoidal Function.id lem : Monoidal→Kleisli Kleisli→Monoidal Function.id
lem = {!!} -- verso-recto Monoidal≃Kleisli lem = {!!} -- verso-recto Monoidal≃Kleisli
t : { : Level} {A B : Set } {a : _ A} {b : B _} t : (§2-fromMonad (Monoidal→Kleisli Kleisli→Monoidal) §2-3.§2.toMonad)
a (Monoidal→Kleisli Kleisli→Monoidal) b a b (§2-fromMonad §2-3.§2.toMonad)
t {a = a} {b} = cong (λ φ a φ b) lem t = cong (λ φ §2-fromMonad (λ{ {ω} φ {{!????!}}}) §2-3.§2.toMonad) {!lem!}
u : { : Level} {A B : Set } {a : _ A} {b : B _} u : (§2-fromMonad (Monoidal→Kleisli Kleisli→Monoidal) §2-3.§2.toMonad) m
{m : _} (a (Monoidal→Kleisli Kleisli→Monoidal) b) m (a b) m (§2-fromMonad §2-3.§2.toMonad) m
u {m = m} = cong (λ φ φ m) t u = cong (λ φ φ m) t
backEq : m (back forth) m m backEq : m (back forth) m m
backEq m = begin backEq m = begin

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@ -1,9 +1,41 @@
{-# OPTIONS --allow-unsolved-metas #-}
module Cat.Wishlist where module Cat.Wishlist where
open import Level open import Level hiding (suc)
open import Cubical
open import Cubical.NType open import Cubical.NType
open import Data.Nat using (_≤_ ; z≤n ; s≤s) open import Data.Nat using (_≤_ ; z≤n ; s≤s ; zero ; suc)
open import Agda.Builtin.Sigma
open import Cubical.NType.Properties open import Cubical.NType.Properties
postulate ntypeCommulative : { n m} {A : Set } n m HasLevel n ⟩₋₂ A HasLevel m ⟩₋₂ A step : {} {A : Set } isContr A (x y : A) isContr (x y)
step (a , contr) x y = {!p , c!}
-- where
-- p : x ≡ y
-- p = begin
-- x ≡⟨ sym (contr x) ⟩
-- a ≡⟨ contr y ⟩
-- y ∎
-- c : (q : x ≡ y) → p ≡ q
-- c q i j = contr (p {!!}) {!!}
-- Contractible types have any given homotopy level.
contrInitial : { : Level} {A : Set } n isContr A HasLevel n A
contrInitial ⟨-2⟩ contr = contr
-- lem' (S ⟨-2⟩) (a , contr) = {!step!}
contrInitial (S ⟨-2⟩) (a , contr) x y = begin
x ≡⟨ sym (contr x)
a ≡⟨ contr y
y
contrInitial (S (S n)) contr x y = {!lvl!} -- Why is this not well-founded?
where
c : isContr (x y)
c = step contr x y
lvl : HasLevel (S n) (x y)
lvl = contrInitial {A = x y} (S n) c
module _ { : Level} {A : Set } where
ntypeCommulative : {n m} n m HasLevel n ⟩₋₂ A HasLevel m ⟩₋₂ A
ntypeCommulative {n = zero} {m} z≤n lvl = {!contrInitial ⟨ m ⟩₋₂ lvl!}
ntypeCommulative {n = .(suc _)} {.(suc _)} (s≤s x) lvl = {!!}