From 896e0d3d37dd60f55dc7ab86023a2ed11e0eb8c9 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Frederik=20Hangh=C3=B8j=20Iversen?= <fhi.1990@gmail.com>
Date: Tue, 13 Mar 2018 10:24:50 +0100
Subject: [PATCH] Stuff about univalence for the category of functors

---
 src/Cat/Categories/Fun.agda | 90 ++++++++++++++++++++++++++++++++++++-
 1 file changed, 88 insertions(+), 2 deletions(-)

diff --git a/src/Cat/Categories/Fun.agda b/src/Cat/Categories/Fun.agda
index 9c1b380..019e8b6 100644
--- a/src/Cat/Categories/Fun.agda
+++ b/src/Cat/Categories/Fun.agda
@@ -6,6 +6,7 @@ open import Data.Product
 
 
 open import Cubical
+open import Cubical.GradLemma
 open import Cubical.NType.Properties
 
 open import Cat.Category
@@ -13,10 +14,10 @@ open import Cat.Category.Functor hiding (identity)
 open import Cat.Category.NaturalTransformation
 
 module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
-  open Category using (Object ; 𝟙)
   module NT = NaturalTransformation ℂ 𝔻
   open NT public
   private
+    module ℂ = Category ℂ
     module 𝔻 = Category 𝔻
   private
     module _ {A B C D : Functor ℂ 𝔻} {θ' : NaturalTransformation A B}
@@ -65,13 +66,98 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
     ; _∘_ = λ {F G H} → NT[_∘_] {F} {G} {H}
     }
 
+  open RawCategory RawFun
+  open Univalence  RawFun
+  module _ {A B : Functor ℂ 𝔻} where
+    module A = Functor A
+    module B = Functor B
+    module _ (p : A ≡ B) where
+      omapP : A.omap ≡ B.omap
+      omapP i = Functor.omap (p i)
+
+      coerceAB : ∀ {X} → 𝔻 [ A.omap X , A.omap X ] ≡ 𝔻 [ A.omap X , B.omap X ]
+      coerceAB {X} = cong (λ φ → 𝔻 [ A.omap X , φ X ]) omapP
+
+      -- The transformation will be the identity on 𝔻. Such an arrow has the
+      -- type `A.omap A → A.omap A`. Which we can coerce to have the type
+      -- `A.omap → B.omap` since `A` and `B` are equal.
+      coe𝟙 : Transformation A B
+      coe𝟙 X = coe coerceAB 𝔻.𝟙
+
+      module _ {a b : ℂ.Object} (f : ℂ [ a , b ]) where
+        nat' : 𝔻 [ coe𝟙 b ∘ A.fmap f ] ≡ 𝔻 [ B.fmap f ∘ coe𝟙 a ]
+        nat' = begin
+          (𝔻 [ coe𝟙 b ∘ A.fmap f ]) ≡⟨ {!!} ⟩
+          (𝔻 [ B.fmap f ∘ coe𝟙 a ]) ∎
+
+      transs : (i : I) → Transformation A (p i)
+      transs = {!!}
+
+      natt : (i : I) → Natural A (p i) {!!}
+      natt = {!!}
+
+      t : Natural A B coe𝟙
+      t = coe c (identityNatural A)
+        where
+        c : Natural A A (identityTrans A) ≡ Natural A B coe𝟙
+        c = begin
+          Natural A A (identityTrans A) ≡⟨ (λ x → {!natt ?!}) ⟩
+          Natural A B coe𝟙 ∎
+        -- cong (λ φ → {!Natural A A (identityTrans A)!}) {!!}
+
+      k : Natural A A (identityTrans A) → Natural A B coe𝟙
+      k n {a} {b} f = res
+        where
+        res : (𝔻 [ coe𝟙 b ∘ A.fmap f ]) ≡ (𝔻 [ B.fmap f ∘ coe𝟙 a ])
+        res = {!!}
+
+      nat : Natural A B coe𝟙
+      nat = nat'
+
+      fromEq : NaturalTransformation A B
+      fromEq = coe𝟙 , nat
+
+  module _ {A B : Functor ℂ 𝔻} where
+    obverse : A ≡ B → A ≅ B
+    obverse p = res
+      where
+      ob  : Arrow A B
+      ob = fromEq p
+      re : Arrow B A
+      re = fromEq (sym p)
+      vr : _∘_ {A = A} {B} {A} re ob ≡ 𝟙 {A}
+      vr = {!!}
+      rv : _∘_ {A = B} {A} {B} ob re ≡ 𝟙 {B}
+      rv = {!!}
+      isInverse : IsInverseOf {A} {B} ob re
+      isInverse = vr , rv
+      iso : Isomorphism {A} {B} ob
+      iso = re , isInverse
+      res : A ≅ B
+      res = ob , iso
+
+    reverse : A ≅ B → A ≡ B
+    reverse iso = {!!}
+
+    ve-re : (y : A ≅ B) → obverse (reverse y) ≡ y
+    ve-re = {!!}
+
+    re-ve : (x : A ≡ B) → reverse (obverse x) ≡ x
+    re-ve = {!!}
+
+    done : isEquiv (A ≡ B) (A ≅ B) (id-to-iso (λ { {A} {B} → isIdentity {A} {B}}) A B)
+    done = {!gradLemma obverse reverse ve-re re-ve!}
+
+  univalent : Univalent (λ{ {A} {B} → isIdentity {A} {B}})
+  univalent = done
+
   instance
     isCategory : IsCategory RawFun
     isCategory = record
       { isAssociative = λ {A B C D} → isAssociative {A} {B} {C} {D}
       ; isIdentity = λ {A B} → isIdentity {A} {B}
       ; arrowsAreSets = λ {F} {G} → naturalTransformationIsSet {F} {G}
-      ; univalent = {!!}
+      ; univalent = univalent
       }
 
   Fun : Category (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') (ℓc ⊔ ℓc' ⊔ ℓd')