Add another approach for univalence in Set

src/Cat/Categories/Sets.agda

@@ -1,11 +1,11 @@
 -- | The category of homotopy sets
-{-# OPTIONS --allow-unsolved-metas --cubical #-}
+{-# OPTIONS --allow-unsolved-metas --cubical --caching #-}
 module Cat.Categories.Sets where
 
 open import Cat.Prelude hiding (_≃_)
 import Data.Product
 
-open import Function using (_∘_)
+open import Function using (_∘_ ; _∘′_)
 
 open import Cubical.Univalence using (univalence ; con ; _≃_ ; idtoeqv ; ua)
 
@@ -267,12 +267,12 @@ module _ (ℓ : Level) where
         res = fwd , bwd , record { verso-recto = refl ; recto-verso = refl }
       conclusion : (hA ≅ hB) ≃ (hA ≡ hB)
       conclusion = trivial? ⊙ step0 ⊙ step1 ⊙ step2
-      t : (hA ≡ hB) ≃ (hA ≅ hB)
-      t = sym≃ conclusion
+      thierry : (hA ≡ hB) ≃ (hA ≅ hB)
+      thierry = sym≃ conclusion
       -- TODO Is the morphism `(_≃_.eqv conclusion)` the same as
       -- `(id-to-iso (λ {A} {B} → isIdentity {A} {B}) hA hB)` ?
-      res : isEquiv (hA ≡ hB) (hA ≅ hB) (_≃_.eqv t)
-      res = _≃_.isEqv t
+      res : isEquiv (hA ≡ hB) (hA ≅ hB) _
+      res = _≃_.isEqv thierry
       thr : (hA ≡ hB) ≃ (hA ≅ hB)
       thr = con _ res
       -- p : _ → (hX : Object) → Path (hA ≅ hB) (hA ≡ hB)
@@ -297,8 +297,51 @@ module _ (ℓ : Level) where
       --
       -- is contractible, which implies univalence.
 
-    univalent[Contr] : ∀ hA → isContr (Σ[ hB ∈ Object ] hA ≅ hB)
-    univalent[Contr] hA = {!!} , {!!}
+    module _ (hA : Object) where
+      open Σ hA renaming (proj₁ to A)
+
+      center : Σ[ hB ∈ Object ] (hA ≅ hB)
+      center = hA , idIso hA
+      open Σ center renaming ({-proj₁ to hA ;-} proj₂ to isoA) using ()
+
+      module _ (y : Σ[ hC ∈ Object ] (hA ≅ hC)) where
+        open Σ y renaming (proj₁ to hC ; proj₂ to hA≅hC)
+        open Σ hC renaming (proj₁ to C)
+
+        open Σ hA≅hC  renaming (proj₁ to obv ; proj₂ to iso)
+        open Σ iso    renaming (proj₁ to inv ; proj₂ to areInv)
+
+        -- Idea:
+        -- Have : hA ≅ hC
+        -- Can I then construct `A Eqv.≅ C`
+        -- Cuz then it follows from univalence
+        A≡C : A ≡ C
+        A≡C = ua s
+          where
+          s0 : A Eqv.≅ C
+          s0 = obv , inv , Eqv.toAreInverses areInv
+          s : A ≃ C
+          s = fromIsomorphism s0
+
+        pObj : hA ≡ hC
+        pObj = lemSig (λ _ → isSetIsProp) hA hC A≡C
+
+        abstract
+          isoEq
+            : (λ i → Σ (A → proj₁ (pObj i)) (Isomorphism {A = hA} {pObj i}))
+            [ idIso hA ≡ hA≅hC ]
+          isoEq = {!!}
+            where
+            d : ∀ iso → (λ _ → Σ (A → A) (Isomorphism {A = hA} {hA}))
+              [ idIso hA ≡ iso ]
+            d iso = {!!}
+
+        isContractible : (hA , idIso hA) ≡ (hC , hA≅hC)
+        isContractible = Σ≡ pObj {!isoEq!}
+        -- isContractible = lemSig prop≅ center y pObj
+
+      univalent[Contr] : isContr (Σ[ hB ∈ Object ] hA ≅ hB)
+      univalent[Contr] = center , isContractible
 
     univalent : Univalent
     univalent = from[Contr] univalent[Contr]

src/Cat/Equivalence.agda

@@ -27,6 +27,16 @@ module _ {ℓa ℓb : Level} where
     Isomorphism : (f : A → B) → Set _
     Isomorphism f = Σ (B → A) λ g → AreInverses f g
 
+    module _ {f : A → B} {g : B → A}
+        (inv : (g ∘ f) ≡ idFun A
+             × (f ∘ g) ≡ idFun B) where
+      open Σ inv renaming (fst to ve-re ; snd to re-ve)
+      toAreInverses : AreInverses f g
+      toAreInverses = record
+        { verso-recto = ve-re
+        ; recto-verso = re-ve
+        }
+
   _≅_ : Set ℓa → Set ℓb → Set _
   A ≅ B = Σ (A → B) Isomorphism