Cosmetics

README.md

@@ -5,11 +5,14 @@ consisting of the proposal for the thesis).
 
 Installation
 ============
-You probably need a very recent version of the Agda compiler. At the time
-of writing the solution has been tested with Agda version 2.6.0-5d84754.
 
 Dependencies
 ------------
+To succesfully compile the following is needed:
+
+* Agda version >= `707ce6042b6a3bdb26521f3fe8dfe5d8a8470a43`.
+* Agda Standard Library >= `87d28d7d753f73abd20665d7bbb88f9d72ed88aa`.
+
 I've used git submodules to manage dependencies. Unfortunately Agda does not
 allow specifying libraries to be used only as local dependencies.
 

src/Cat/Category.agda

@@ -99,7 +99,8 @@ record IsCategory {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) : Set (lsuc
   id-to-iso : (A B : Object) → A ≡ B → A ≅ B
   id-to-iso A B eq = transp (\ i → A ≅ eq i) (idIso A)
 
-  -- TODO: might want to implement isEquiv differently, there are 3
+  -- TODO: might want to implement isEquiv
+  -- differently, there are 3
   -- equivalent formulations in the book.
   Univalent : Set (ℓa ⊔ ℓb)
   Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≅ B) (id-to-iso A B)
@@ -144,20 +145,22 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
     -- Why choose `x`'s `propIsAssociative`?
     -- Well, probably it could be pulled out of the record.
     { assoc = x.propIsAssociative x.assoc y.assoc i
-    ; ident = x.propIsIdentity x.ident y.ident i
+    ; ident = ident' i
     ; arrowIsSet = x.propArrowIsSet x.arrowIsSet y.arrowIsSet i
     ; univalent = eqUni i
     }
     where
       module x = IsCategory x
       module y = IsCategory y
+      ident' = x.propIsIdentity x.ident y.ident
+      ident'' = ident' i
       xuni : x.Univalent
       xuni = x.univalent
       yuni : y.Univalent
       yuni = y.univalent
       open RawCategory ℂ
-      T :  I → Set (ℓa ⊔ ℓb)
+      Pp : (x.ident ≡ y.ident) → I → Set (ℓa ⊔ ℓb)
-      T i = {A B : Object} →
+      Pp eqIdent i = {A B : Object} →
         isEquiv (A ≡ B) (A x.≅ B)
           (λ A≡B →
             transp
@@ -166,17 +169,21 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
             (λ f → Σ-syntax (Arrow (A≡B j) A) (λ g → g ∘ f ≡ 𝟙 × f ∘ g ≡ 𝟙)))
             ( 𝟙
             , 𝟙
-            , x.propIsIdentity x.ident y.ident i
+            , ident' i
             )
           )
+      T : I → Set (ℓa ⊔ ℓb)
+      T = Pp {!ident'!}
       open Cubical.NType.Properties
       test : (λ _ → x.Univalent) [ xuni ≡ xuni ]
       test = refl
       t = {!!}
       P : (uni : x.Univalent) → xuni ≡ uni → Set (ℓa ⊔ ℓb)
       P = {!!}
+      -- T i0 ≡ x.Univalent
+      -- T i1 ≡ y.Univalent
       eqUni : T [ xuni ≡ yuni ]
-      eqUni = pathJprop {x = x.Univalent} P {!!} i
+      eqUni = {!!}
 
 
 record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where

src/Cat/Category/Functor.agda

@@ -47,7 +47,6 @@ module _ {ℓc ℓc' ℓd ℓd'}
 open IsFunctor
 open Functor
 
--- TODO: Is `IsFunctor` a proposition?
 module _
     {ℓa ℓb : Level}
     {ℂ 𝔻 : Category ℓa ℓb}
@@ -56,11 +55,8 @@ module _
   private
     module 𝔻 = IsCategory (isCategory 𝔻)
 
-  -- isProp  : Set ℓ
+  propIsFunctor : isProp (IsFunctor _ _ F)
-  -- isProp  = (x y : A) → x ≡ y
+  propIsFunctor isF0 isF1 i = record
-
-  IsFunctorIsProp : isProp (IsFunctor _ _ F)
-  IsFunctorIsProp isF0 isF1 i = record
     { ident = 𝔻.arrowIsSet _ _ isF0.ident isF1.ident i
     ; distrib = 𝔻.arrowIsSet _ _ isF0.distrib isF1.distrib i
     }
@@ -81,7 +77,7 @@ module _
 
   IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
   IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
-    (\ F → IsFunctorIsProp {F = F}) (\ i → F i)
+    (\ F → propIsFunctor {F = F}) (\ i → F i)
     where
       open import Cubical.NType.Properties using (lemPropF)