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Rely on global cubical again

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Frederik Hanghøj Iversen 2018-02-16 11:36:44 +01:00
parent 8a3a519955
commit 23c458983c
6 changed files with 51 additions and 51 deletions
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8
cat.agda-lib
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@ -2,13 +2,11 @@ name: cat
-- version: 0.0.1
-- description:
-- A formalization of category theory in Agda using cubical type theory.
-- depend:
-- standard-library
-- cubical
depend:
standard-library
cubical
include:
src
libs/agda-stdlib/src
libs/cubical/src
-- libraries:
-- libs/agda-stdlib
-- libs/cubical

2
libs/agda-stdlib

@ -1 +1 @@
Subproject commit b5bfbc3c170b0bd0c9aaac1b4d4b3f9b06832bf6
Subproject commit 157497a5335ad0069c7aaffbc65932c40a28ee68

2
libs/cubical

@ -1 +1 @@
Subproject commit 1d6730c4999daa2b04e9dd39faa0791d0b5c3b48
Subproject commit 12c2c628e9e202f1698a4c32e0356d5ca8cb6151

86
src/Cat/Categories/Fun.agda
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@ -7,6 +7,7 @@ open import Function
open import Data.Product
import Cubical.GradLemma
module UIP = Cubical.GradLemma
open import Cubical.Sigma
open import Cat.Category
open import Cat.Category.Functor
@ -118,48 +119,49 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
lem : (λ _ → Natural F G θ) [ (λ f → θNat f) ≡ (λ f → θNat' f) ]
lem = λ i f → 𝔻.arrowIsSet _ _ (θNat f) (θNat' f) i
naturalTransformationIsSets f : isSet (NaturalTransformation F G)
f a b p q i = res
where
k : (θ : Transformation F G) → (xx yy : Natural F G θ) → xx ≡ yy
k θ x y = let kk = naturalIsProp θ x y in {!!}
res : a ≡ b
res j = {!!} , {!!}
-- naturalTransformationIsSets σa σb p q
-- where
-- -- In Andrea's proof `lemSig` he proves something very similiar to
-- -- what I'm doing here, just for `Cubical.FromPathPrelude.Σ` rather
-- -- than `Σ`. In that proof, he just needs *one* proof that the first
-- -- components are equal - hence the arbitrary usage of `p` here.
-- secretSauce : proj₁ σa ≡ proj₁ σb
-- secretSauce i = proj₁ (p i)
-- lemSig : σa ≡ σb
-- lemSig i = (secretSauce i) , (UIP.lemPropF naturalIsProp secretSauce) {proj₂ σa} {proj₂ σb} i
-- res : p ≡ q
-- res = {!!}
naturalTransformationIsSets (θ , θNat) (η , ηNat) p q i j
= θ-η
-- `i or `j - `p'` or `q'`?
, {!!} -- UIP.lemPropF {B = Natural F G} (λ x → {!!}) {(θ , θNat)} {(η , ηNat)} {!!} i
-- naturalIsSet i (λ i → {!!} i) {!!} {!!} i j
-- naturalIsSet {!p''!} {!p''!} {!!} i j
-- λ f k → 𝔻.arrowIsSet (λ l → proj₂ (p l) f k) (λ l → proj₂ (p l) f k) {!!} {!!}
where
θ≡η θ≡η' : θ ≡ η
θ≡η i = proj₁ (p i)
θ≡η' i = proj₁ (q i)
θ-η : Transformation F G
θ-η = transformationIsSet _ _ θ≡η θ≡η' i j
θNat≡ηNat : (λ i → Natural F G (θ≡η i)) [ θNat ≡ ηNat ]
θNat≡ηNat i = proj₂ (p i)
θNat≡ηNat' : (λ i → Natural F G (θ≡η' i)) [ θNat ≡ ηNat ]
θNat≡ηNat' i = proj₂ (q i)
k : Natural F G (θ≡η i)
k = θNat≡ηNat i
k' : Natural F G (θ≡η' i)
k' = θNat≡ηNat' i
t : Natural F G θ-η
t = naturalIsProp {!θ!} {!!} {!!} {!!}
naturalTransformationIsSets : isSet (NaturalTransformation F G)
naturalTransformationIsSets = {!sigPresSet!}
-- f a b p q i = res
-- where
-- k : (θ : Transformation F G) → (xx yy : Natural F G θ) → xx ≡ yy
-- k θ x y = let kk = naturalIsProp θ x y in {!!}
-- res : a ≡ b
-- res j = {!!} , {!!}
-- -- naturalTransformationIsSets σa σb p q
-- -- where
-- -- -- In Andrea's proof `lemSig` he proves something very similiar to
-- -- -- what I'm doing here, just for `Cubical.FromPathPrelude.Σ` rather
-- -- -- than `Σ`. In that proof, he just needs *one* proof that the first
-- -- -- components are equal - hence the arbitrary usage of `p` here.
-- -- secretSauce : proj₁ σa ≡ proj₁ σb
-- -- secretSauce i = proj₁ (p i)
-- -- lemSig : σa ≡ σb
-- -- lemSig i = (secretSauce i) , (UIP.lemPropF naturalIsProp secretSauce) {proj₂ σa} {proj₂ σb} i
-- -- res : p ≡ q
-- -- res = {!!}
-- naturalTransformationIsSets (θ , θNat) (η , ηNat) p q i j
-- = θ-η
-- -- `i or `j - `p'` or `q'`?
-- , {!!} -- UIP.lemPropF {B = Natural F G} (λ x → {!!}) {(θ , θNat)} {(η , ηNat)} {!!} i
-- -- naturalIsSet i (λ i → {!!} i) {!!} {!!} i j
-- -- naturalIsSet {!p''!} {!p''!} {!!} i j
-- -- λ f k → 𝔻.arrowIsSet (λ l → proj₂ (p l) f k) (λ l → proj₂ (p l) f k) {!!} {!!}
-- where
-- θ≡η θ≡η' : θ ≡ η
-- θ≡η i = proj₁ (p i)
-- θ≡η' i = proj₁ (q i)
-- θ-η : Transformation F G
-- θ-η = transformationIsSet _ _ θ≡η θ≡η' i j
-- θNat≡ηNat : (λ i → Natural F G (θ≡η i)) [ θNat ≡ ηNat ]
-- θNat≡ηNat i = proj₂ (p i)
-- θNat≡ηNat' : (λ i → Natural F G (θ≡η' i)) [ θNat ≡ ηNat ]
-- θNat≡ηNat' i = proj₂ (q i)
-- k : Natural F G (θ≡η i)
-- k = θNat≡ηNat i
-- k' : Natural F G (θ≡η' i)
-- k' = θNat≡ηNat' i
-- t : Natural F G θ-η
-- t = naturalIsProp {!θ!} {!!} {!!} {!!}
module _ {A B C D : Functor ℂ 𝔻} {θ' : NaturalTransformation A B}
{η' : NaturalTransformation B C} {ζ' : NaturalTransformation C D} where

2
src/Cat/Category.agda
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@ -12,7 +12,7 @@ open import Data.Product renaming
open import Data.Empty
import Function
open import Cubical
open import Cubical.GradLemma using ( propIsEquiv )
open import Cubical.NType.Properties using ( propIsEquiv )
∃! : ∀ {a b} {A : Set a}
→ (A → Set b) → Set (a ⊔ b)

2
src/Cat/Category/Functor.agda
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@ -83,7 +83,7 @@ module _
IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
(\ F → IsFunctorIsProp {F = F}) (\ i → F i)
where
open import Cubical.GradLemma using (lemPropF)
open import Cubical.NType.Properties using (lemPropF)
module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
Functor≡ : {F G : Functor ℂ 𝔻}