diff --git a/cat.agda-lib b/cat.agda-lib
index cf8593a..1b73d48 100644
--- a/cat.agda-lib
+++ b/cat.agda-lib
@@ -1,8 +1,7 @@
 name: cat
-include:
-  src
-  --libs/cubical-demo
+-- description:
+--   A formalization of category theory in Agda using cubical type theory.
 depend:
-  agda-prelude
   standard-library
-  --cubical-demo
+  cubical
+include: src
diff --git a/libraries b/libraries
new file mode 100644
index 0000000..1813ad5
--- /dev/null
+++ b/libraries
@@ -0,0 +1 @@
+libs/cubical-demo
diff --git a/src/Cat.agda b/src/Cat.agda
deleted file mode 100644
index bc43564..0000000
--- a/src/Cat.agda
+++ /dev/null
@@ -1,150 +0,0 @@
-{-# OPTIONS --cubical #-}
-module Cat where
-
-open import Agda.Primitive
-open import Prelude.Function
-open import PathPrelude
-open ≡-Reasoning
-
--- Questions:
---
--- Is there a canonical `postulate trustme : {l : Level} {A : Set l} -> A`
--- in Agda (like undefined in Haskell).
---
--- Does Agda have a way to specify local submodules?
---
--- Can I open parts of a record, like say:
---
---     open Functor {{...}} ( fmap )
-
-record Functor {a b} (F : Set a → Set b) : Set (lsuc a ⊔ b) where
-  field
-    -- morphisms
-    fmap : ∀ {A B} → (A → B) → F A → F B
-    -- laws
-    identity : { A : Set a }
-      -> id ≡ fmap (id { A = A })
-    fusion : { A B C : Set a } ( f : B -> C ) ( g : A -> B )
-      -> fmap f ∘ fmap g ≡ fmap (f ∘ g)
-
-open Functor {{ ... }} hiding ( identity )
-
-_<$>_ : {a b : Level} {F : Set a → Set b} {{r : Functor F}} {A B : Set a} → (A → B) → F A → F B
-_<$>_ = fmap
-
-record Identity (A : Set) : Set where
-  constructor identity
-  field
-    runIdentity : A
-
-open Identity {{...}}
-
-instance
-  IdentityFunctor : Functor Identity
-  IdentityFunctor = record
-    { fmap = λ { f (identity a) -> identity (f a) }
-    ; identity = refl
-    ; fusion = λ { f g -> refl }
-    }
-
-data Maybe ( A : Set ) : Set where
-  nothing : Maybe A
-  just    : A -> Maybe A
-
-instance
-  MaybeFunctor : Functor Maybe
-  Functor.fmap MaybeFunctor f nothing = nothing
-  Functor.fmap MaybeFunctor f (just x) = just (f x)
-  Functor.identity MaybeFunctor = fun-ext (λ { nothing → refl ; (just x) → refl })
-  Functor.fusion MaybeFunctor f g = fun-ext (λ
-    -- QUESTION: Why does inlining `lem₀` give rise to an error?
-    { nothing → lem₀
-    ; (just x) → {! refl!}
-{-    ; (just x) → begin
-      (fmap f ∘ fmap g) (just x)  ≡⟨⟩
-      (fmap f (fmap g (just x)))  ≡⟨⟩
-      (fmap f (just (g x)))       ≡⟨⟩
-      (just (f (g x)))            ≡⟨⟩
-      just ((f ∘ g) x)              ∎-}
-    })
-    where
-      lem₀ : nothing ≡ nothing
-      lem₀ = refl
-      -- `lem₁` and `fusionjust` are the same.
-      lem₁ : {A B C : Set} {f : B → C} {g : A → B} {x : A}
-        → just (f (g x)) ≡ just (f (g x))
-      lem₁ = refl
-      fusionjust : { A B C : Set } { x : A } { f : B -> C } { g : A -> B }
-        -> (fmap f ∘ fmap g) (just x) ≡ just ((f ∘ g) x)
-      fusionjust {x = x} {f = f} {g = g} = begin
-        (fmap f ∘ fmap g) (just x)  ≡⟨⟩
-        (fmap f (fmap g (just x)))  ≡⟨⟩
-        (fmap f (just (g x)))       ≡⟨⟩
-        (just (f (g x)))            ≡⟨⟩
-        just ((f ∘ g) x)              ∎
-
-
-record Applicative {a} (F : Set a -> Set a) : Set (lsuc a) where
-  field
-    -- morphisms
-    pure : { A : Set a } -> A -> F A
-    ap   : { A B : Set a } -> F (A -> B) -> F A -> F B
-    -- laws
-    -- `ap-identity` is paraphrased from the documentation for Applicative on hackage.
-    ap-identity : { A : Set a } -> ap (pure (id { A = A })) ≡ id
-    composition : { A B C : Set a } ( u : F (B -> C) ) ( v : F (A -> B) ) ( w : F A )
-      -> ap (ap (ap (pure _∘′_) u) v) w ≡ ap u (ap v w)
-    homomorphism : { A B : Set a } { a : A } { f : A -> B }
-      -> ap (pure f) (pure a) ≡ pure (f a)
-    interchange : { A B : Set a } ( u : F (A -> B) ) ( a : A )
-      -> ap u (pure a) ≡ ap (pure (_$ a)) u
-
-open Applicative {{...}}
-_<*>_ = ap
-
-instance
-  IdentityApplicative : Applicative Identity
-  Applicative.pure IdentityApplicative = identity
-  Applicative.ap IdentityApplicative (identity f) (identity a) = identity (f a)
-  Applicative.ap-identity IdentityApplicative = refl
-  Applicative.composition IdentityApplicative _ _ _ = refl
-  Applicative.homomorphism IdentityApplicative = refl
-  Applicative.interchange IdentityApplicative _ _ = refl
-
--- QUESTION: Is this provable? How?
--- I suppose this would be `≡`'s functor-instance.
-equiv-app : {A B : Set} -> {f g : A -> B} -> {a : A} -> f ≡ g -> f a ≡ g a
-equiv-app f≡g = {!!}
-
-instance
-  MaybeApplicative : Applicative Maybe
-  Applicative.pure MaybeApplicative = just
-  Applicative.ap MaybeApplicative nothing _ = nothing
-  Applicative.ap MaybeApplicative (just f) x = fmap f x
-  -- QUESTION: Why does termination-check fail when I use `refl` in both cases below?
-  Applicative.ap-identity MaybeApplicative = fun-ext (λ
-    { nothing → {!refl!}
-    ; (just x) → {!refl!}
-    })
-  Applicative.composition MaybeApplicative nothing v w = refl
-  Applicative.composition MaybeApplicative (just x) nothing w = refl
-  -- Easy-mode:
-  --  Applicative.composition MaybeApplicative {u = just f} {just g} {nothing} = refl
-  --  Applicative.composition MaybeApplicative {u = just f} {just g} {just x} = refl
-  -- Re-use mode:
-  -- QUESTION: What's wrong here?
-  Applicative.composition MaybeApplicative (just f) (just g) w = sym (equiv-app lem)
-    where
-    lem : ∀ {A B C} {f : B → C} {g : A → B} {w : Maybe A} →
-      (fmap f) ∘ (fmap g) ≡
-      (Functor.fmap MaybeFunctor (f ∘ g))
-    lem = Functor.fusion MaybeFunctor _ _
-{-
-  Applicative.composition MaybeApplicative { u = u } { v = v } { w = w } = begin
-    ap (ap (ap (pure _∘′_) u) v) w ≡⟨⟩
-    ap (ap (fmap _∘′_      u) v) w ≡⟨ {!!} ⟩
-    {!!} ∎
--}
-  Applicative.homomorphism MaybeApplicative = refl
-  Applicative.interchange MaybeApplicative nothing a = refl
-  Applicative.interchange MaybeApplicative (just x) a = refl
diff --git a/src/Category.agda b/src/Category.agda
new file mode 100644
index 0000000..ef5c2c4
--- /dev/null
+++ b/src/Category.agda
@@ -0,0 +1,196 @@
+{-# OPTIONS --cubical #-}
+
+module Category where
+
+open import Agda.Primitive
+open import Data.Unit.Base
+open import Data.Product
+open import Cubical.PathPrelude
+
+postulate undefined : {ℓ : Level} → {A : Set ℓ} → A
+
+record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
+  field
+    Object : Set ℓ
+    Arrow  : Object → Object → Set ℓ'
+    𝟙      : {o : Object} → Arrow o o
+    _⊕_    : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c
+    assoc : { A B C D : Object } { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
+      → h ⊕ (g ⊕ f) ≡ (h ⊕ g) ⊕ f
+    ident  : { A B : Object } { f : Arrow A B }
+      → f ⊕ 𝟙 ≡ f × 𝟙 ⊕ f ≡ f
+  infixl 45 _⊕_
+  dom : { a b : Object } → Arrow a b → Object
+  dom {a = a} _ = a
+  cod : { a b : Object } → Arrow a b → Object
+  cod {b = b} _ = b
+
+open Category public
+
+record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Category {ℓd} {ℓd'})
+  : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
+  private
+    open module C = Category C
+    open module D = Category D
+  field
+    F : C.Object → D.Object
+    f : {c c' : C.Object} → C.Arrow c c' → D.Arrow (F c) (F c')
+    ident   : { c : C.Object } → f (C.𝟙 {c}) ≡ D.𝟙 {F c}
+    -- TODO: Avoid use of ugly explicit arguments somehow.
+    -- This guy managed to do it:
+    --    https://github.com/copumpkin/categories/blob/master/Categories/Functor/Core.agda
+    distrib : { c c' c'' : C.Object} {a : C.Arrow c c'} {a' : C.Arrow c' c''}
+      → f (a' C.⊕ a) ≡ f a' D.⊕ f a
+
+FunctorComp : ∀ {ℓ ℓ'} {a b c : Category {ℓ} {ℓ'}} → Functor b c → Functor a b → Functor a c
+FunctorComp {a = a} {b = b} {c = c} F G =
+  record
+    { F = F.F ∘ G.F
+    ; f = F.f ∘ G.f
+    ; ident = λ { {c = obj} →
+      let --t : (F.f ∘ G.f) (𝟙 a) ≡ (𝟙 c)
+          g-ident = G.ident
+          k : F.f (G.f {c' = obj} (𝟙 a)) ≡ F.f (G.f (𝟙 a))
+          k = refl {x = F.f (G.f (𝟙 a))}
+          t : F.f (G.f (𝟙 a)) ≡ (𝟙 c)
+          -- t = subst F.ident (subst G.ident k)
+          t = undefined
+      in t }
+    ; distrib = undefined -- subst F.distrib (subst G.distrib refl)
+    }
+    where
+      open module F = Functor F
+      open module G = Functor G
+
+-- The identity functor
+Identity : {ℓ ℓ' : Level} → {C : Category {ℓ} {ℓ'}} → Functor C C
+Identity = record { F = λ x → x ; f = λ x → x ; ident = refl ; distrib = refl }
+
+module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} { A B : Object ℂ } where
+  private
+    open module ℂ = Category ℂ
+    _+_ = ℂ._⊕_
+
+  Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
+  Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g + f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
+
+  Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
+  Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
+
+  Monomorphism : {X : ℂ.Object} → (f : ℂ.Arrow A B) → Set ℓ'
+  Monomorphism {X} f = ( g₀ g₁ : ℂ.Arrow X A ) → f + g₀ ≡ f + g₁ → g₀ ≡ g₁
+
+  iso-is-epi : ∀ {X} (f : ℂ.Arrow A B) → Isomorphism f → Epimorphism {X = X} f
+  -- Idea: Pre-compose with f- on both sides of the equality of eq to get
+  -- g₀ + f + f- ≡ g₁ + f + f-
+  -- which by left-inv reduces to the goal.
+  iso-is-epi f (f- , left-inv , right-inv) g₀ g₁ eq =
+     trans (sym (fst ℂ.ident))
+       ( trans (cong (_+_ g₀) (sym right-inv))
+         ( trans ℂ.assoc
+           ( trans (cong (λ x → x + f-) eq)
+             ( trans (sym ℂ.assoc)
+               ( trans (cong (_+_ g₁) right-inv) (fst ℂ.ident))
+             )
+           )
+         )
+       )
+
+  iso-is-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Monomorphism {X = X} f
+  -- For the next goal we do something similar: Post-compose with f- and use
+  -- right-inv to get the goal.
+  iso-is-mono f (f- , (left-inv , right-inv)) g₀ g₁ eq =
+    trans (sym (snd ℂ.ident))
+      ( trans (cong (λ x → x + g₀) (sym left-inv))
+        ( trans (sym ℂ.assoc)
+          ( trans (cong (_+_ f-) eq)
+            ( trans ℂ.assoc
+              ( trans (cong (λ x → x + g₁) left-inv) (snd ℂ.ident)
+              )
+            )
+          )
+        )
+      )
+
+  iso-is-epi-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
+  iso-is-epi-mono f iso = iso-is-epi f iso , iso-is-mono f iso
+
+data ⊥ : Set where
+
+¬_ : {ℓ : Level} → Set ℓ → Set ℓ
+¬ A = A → ⊥
+
+{-
+epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
+epi-mono-is-not-iso f =
+  let k = f {!!} {!!} {!!} {!!}
+  in {!!}
+-}
+
+_≅_ : { ℓ ℓ' : Level } → { ℂ : Category {ℓ} {ℓ'} } → ( A B : Object ℂ ) → Set ℓ'
+_≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ.Arrow A B ] (Isomorphism {ℂ = ℂ} f)
+  where
+    open module ℂ = Category ℂ
+
+Product : {ℓ : Level} → ( C D : Category {ℓ} {ℓ} ) → Category {ℓ} {ℓ}
+Product C D =
+  record
+    { Object = C.Object × D.Object
+    ; Arrow = λ { (c , d) (c' , d') →
+      let carr = C.Arrow c c'
+          darr = D.Arrow d d'
+      in carr × darr}
+    ; 𝟙 = C.𝟙 , D.𝟙
+    ; _⊕_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → bc∈C C.⊕ ab∈C , bc∈D D.⊕ ab∈D}
+    ; assoc = eqpair C.assoc D.assoc
+    ; ident =
+      let (Cl , Cr) = C.ident
+          (Dl , Dr) = D.ident
+      in eqpair Cl Dl , eqpair Cr Dr
+    }
+  where
+    open module C = Category C
+    open module D = Category D
+    -- Two pairs are equal if their components are equal.
+    eqpair : {ℓ : Level} → { A : Set ℓ } → { B : Set ℓ } → { a a' : A } → { b b' : B } → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
+    eqpair {a = a} {b = b} eqa eqb = subst eqa (subst eqb (refl {x = (a , b)}))
+
+Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
+Opposite ℂ =
+  record
+    { Object = ℂ.Object
+    ; Arrow = λ A B → ℂ.Arrow B A
+    ; 𝟙 = ℂ.𝟙
+    ; _⊕_ = λ g f → f ℂ.⊕ g
+    ; assoc = sym ℂ.assoc
+    ; ident = swap ℂ.ident
+    }
+  where
+    open module ℂ = Category ℂ
+
+CatCat : {ℓ ℓ' : Level} → Category {ℓ-suc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
+CatCat {ℓ} {ℓ'} =
+  record
+    { Object = Category {ℓ} {ℓ'}
+    ; Arrow = Functor
+    ; 𝟙 = Identity
+    ; _⊕_ = FunctorComp
+    ; assoc = undefined
+    ; ident = λ { {f = f} →
+      let eq : f ≡ f
+          eq = refl
+      in undefined , undefined}
+    }
+
+Hom : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → (A B : Object ℂ) → Set ℓ'
+Hom {ℂ = ℂ} A B = Arrow ℂ A B
+
+module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
+  private
+    Obj = Object ℂ
+    Arr = Arrow ℂ
+    _+_ = _⊕_ ℂ
+
+  HomFromArrow : (A : Obj) → {B B' : Obj} → (g : Arr B B')
+    → Hom {ℂ = ℂ} A B → Hom {ℂ = ℂ} A B'
+  HomFromArrow _A g = λ f → g + f
diff --git a/src/Category/Bij.agda b/src/Category/Bij.agda
new file mode 100644
index 0000000..217c329
--- /dev/null
+++ b/src/Category/Bij.agda
@@ -0,0 +1,43 @@
+{-# OPTIONS --cubical #-}
+
+open import Cubical.PathPrelude hiding ( Id )
+
+module _ {A : Set} {a : A} {P : A → Set} where
+  Q : A → Set
+  Q a = A
+
+  t : Σ[ a ∈ A ] P a → Q a
+  t (a , Pa) = a
+  u : Q a → Σ[ a ∈ A ] P a
+  u a = a , {!!}
+
+  tu-bij : (a : Q a) → (t ∘ u) a ≡ a
+  tu-bij a = refl
+
+  v : P a → Q a
+  v x = {!!}
+  w : Q a → P a
+  w x = {!!}
+
+  vw-bij : (a : P a) → (w ∘ v) a ≡ a
+  vw-bij a = refl
+  -- tubij a with (t ∘ u) a
+  -- ... | q = {!!}
+
+  data Id {A : Set} (a : A) : Set where
+    id : A → Id a
+
+  data Id' {A : Set} (a : A) : Set where
+    id' : A → Id' a
+
+  T U : Set
+  T = Id a
+  U = Id' a
+
+  f : T → U
+  f (id x) = id' x
+  g : U → T
+  g (id' x) = id x
+
+  fg-bij : (x : U) → (f ∘ g) x ≡ x
+  fg-bij (id' x) = {!!}
diff --git a/src/Category/Free.agda b/src/Category/Free.agda
new file mode 100644
index 0000000..c67f25a
--- /dev/null
+++ b/src/Category/Free.agda
@@ -0,0 +1,38 @@
+module Category.Free where
+
+open import Agda.Primitive
+open import Cubical.PathPrelude hiding (Path)
+
+open import Category as C
+
+module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
+  private
+    open module ℂ = Category ℂ
+    Obj = ℂ.Object
+
+  Path : ( a b : Obj ) → Set
+  Path a b = undefined
+
+  postulate emptyPath : (o : Obj) → Path o o
+
+  postulate concatenate : {a b c : Obj} → Path b c → Path a b → Path a c
+
+  private
+    module _ {A B C D : Obj} {r : Path A B} {q : Path B C} {p : Path C D} where
+      postulate
+        p-assoc : concatenate {A} {C} {D} p (concatenate {A} {B} {C} q r)
+          ≡ concatenate {A} {B} {D} (concatenate {B} {C} {D} p q) r
+    module _ {A B : Obj} {p : Path A B} where
+      postulate
+        ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
+        ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p
+
+  Free : Category
+  Free = record
+    { Object = Obj
+    ; Arrow = Path
+    ; 𝟙 = λ {o} → emptyPath o
+    ; _⊕_ = λ {a b c} → concatenate {a} {b} {c}
+    ; assoc = p-assoc
+    ; ident = ident-r , ident-l
+    }
diff --git a/src/Category/Pathy.agda b/src/Category/Pathy.agda
new file mode 100644
index 0000000..7738726
--- /dev/null
+++ b/src/Category/Pathy.agda
@@ -0,0 +1,52 @@
+{-# OPTIONS --cubical #-}
+
+module Category.Pathy where
+
+open import Cubical.PathPrelude
+
+{-
+module _ {ℓ ℓ'} {A : Set ℓ} {x : A}
+  (P : ∀ y → x ≡ y → Set ℓ') (d : P x ((λ i → x))) where
+  pathJ' : (y : A) → (p : x ≡ y) → P y p
+  pathJ' _ p = transp (λ i → uncurry P (contrSingl p i)) d
+
+  pathJprop' : pathJ' _ refl ≡ d
+  pathJprop' i
+    = primComp (λ _ → P x refl) i (λ {j (i = i1) → d}) d
+
+
+module _ {ℓ ℓ'} {A : Set ℓ}
+  (P : (x y : A) → x ≡ y → Set ℓ') (d : (x : A) → P x x refl) where
+  pathJ'' : (x y : A) → (p : x ≡ y) → P x y p
+  pathJ'' _ _ p = transp (λ i →
+    let
+      P' = uncurry P
+      q  = (contrSingl p i)
+    in
+      {!uncurry (uncurry P)!} ) d
+-}
+
+module _ {ℓ ℓ'} {A : Set ℓ}
+  (C : (x y : A) → x ≡ y → Set ℓ')
+  (c : (x : A) → C x x refl) where
+
+  =-ind : (x y : A) → (p : x ≡ y) → C x y p
+  =-ind x y p = pathJ (C x) (c x) y p
+
+module _ {ℓ ℓ' : Level} {A : Set ℓ} {P : A → Set ℓ} {x y : A} where
+  private
+    D : (x y : A) → (x ≡ y) → Set ℓ
+    D x y p = P x → P y
+
+    id : {ℓ : Level} → {A : Set ℓ} → A → A
+    id x = x
+
+    d : (x : A) → D x x refl
+    d x = id {A = P x}
+
+  -- the p refers to the third argument
+  liftP : x ≡ y → P x → P y
+  liftP p = =-ind D d x y p
+
+  -- lift' : (u : P x) → (p : x ≡ y) → (x , u) ≡ (y , liftP p u)
+  -- lift' u p = {!!}
diff --git a/src/Category/Rel.agda b/src/Category/Rel.agda
new file mode 100644
index 0000000..2091558
--- /dev/null
+++ b/src/Category/Rel.agda
@@ -0,0 +1,159 @@
+{-# OPTIONS --cubical #-}
+module Category.Rel where
+
+open import Data.Product
+open import Cubical.PathPrelude
+open import Cubical.GradLemma
+open import Agda.Primitive
+open import Category
+
+-- Sets are built-in to Agda. The set of all small sets is called Set.
+
+Fun : {ℓ : Level} → ( T U : Set ℓ ) → Set ℓ
+Fun T U = T → U
+
+𝕊et-as-Cat : {ℓ : Level} → Category {lsuc ℓ} {ℓ}
+𝕊et-as-Cat {ℓ} = record
+  { Object = Set ℓ
+  ; Arrow = λ T U → Fun {ℓ} T U
+  ; 𝟙 = λ x → x
+  ; _⊕_  = λ g f x → g ( f x )
+  ; assoc = refl
+  ; ident = funExt (λ x → refl) , funExt (λ x → refl)
+  }
+
+-- Subsets are predicates over some type.
+Subset : {ℓ : Level} → ( A : Set ℓ ) → Set (ℓ ⊔ lsuc lzero)
+Subset A = A → Set
+-- Subset : {ℓ ℓ' : Level} → ( A : Set ℓ ) → Set (ℓ ⊔ lsuc ℓ')
+-- Subset {ℓ' = ℓ'} A = A → Set ℓ'
+-- {a ∈ A | P a}
+
+-- subset-syntax : {ℓ ℓ' : Level} → (A : Set ℓ) → (P : A → Set ℓ') → ( a : A ) → Set ℓ'
+-- subset-syntax A P a = P a
+-- infix 2 subset-syntax
+
+-- syntax subset P a = << a ∈ A >>>
+-- syntax subset P = ⦃ a ∈ A | P a ⦄
+-- syntax subset-syntax A (λ a → B) = ⟨ a foo A ∣ B ⟩
+
+-- Membership is function applicatiom.
+_∈_ : {ℓ : Level} {A : Set ℓ} → A → Subset A → Set
+s ∈ S = S s
+
+infixl 45 _∈_
+
+-- The diagnoal of a set is a synonym for equality.
+Diag : ∀ S → Subset (S × S)
+Diag S (s₀ , s₁) = s₀ ≡ s₁
+-- Diag S = subset (S × S) (λ {(p , q) → p ≡ q})
+-- Diag S = ⟨ ? foo ? ∣ ? ⟩
+-- Diag S (s₀ , s₁) = ⦃ (s₀ , s₁) ∈ S | s₀ ≡ s₁ ⦄
+
+module _ {A B : Set} {S : Subset (A × B)} (ab : A × B) where
+  private
+    a : A
+    a = fst ab
+    b : B
+    b = snd ab
+
+  module _ where
+    private
+      forwards : ((a , b) ∈ S)
+        → (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+      forwards ab∈S = a , (refl , ab∈S)
+
+      backwards : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+        → (a , b) ∈ S
+      backwards (a' , (a=a' , a'b∈S)) = subst (sym a=a') a'b∈S
+
+      isbijective : (x : (a , b) ∈ S) → (backwards ∘ forwards) x ≡ x
+      -- isbijective x = pathJ (λ y x₁ → (backwards ∘ forwards) x ≡ x) {!!} {!!} {!!}
+      isbijective x = pathJprop (λ y _ → y) x
+
+      postulate
+        back-fwd : (x : Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+          → (forwards ∘ backwards) x ≡ x
+      -- back-fwd (a , (p , ab∈S))
+      -- = =-ind (λ x y p → {!(forwards ∘ backwards) x ≡ x!}) {!!} {!!} {!!} p
+      -- = pathJprop (λ y _ → snd (snd y)) ab∈S
+      -- has type P x refl where P is the first argument
+{-
+
+module _ {ℓ ℓ'} {A : Set ℓ} {x : A}
+  (P : ∀ y → x ≡ y → Set ℓ') (d : P x ((λ i → x))) where
+  pathJ : (y : A) → (p : x ≡ y) → P y p
+  pathJ _ p = transp (λ i → uncurry P (contrSingl p i)) d
+
+  pathJprop : pathJ _ refl ≡ d
+  pathJprop i = primComp (λ _ → P x refl) i (λ {j (i = i1) → d}) d
+-}
+      isequiv : isEquiv
+        (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+        ((a , b) ∈ S)
+        backwards
+--      isequiv ab∈S = (forwards ab∈S , sym (isbijective ab∈S)) , λ y → fiberhelp y
+      isequiv y = gradLemma backwards forwards isbijective back-fwd y
+
+      equi : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+        ≃ (a , b) ∈ S
+      equi = backwards , isequiv
+
+    ident-l : (Σ[ a' ∈ A ] (a , a') ∈ Diag A × (a' , b) ∈ S)
+      ≡ (a , b) ∈ S
+    ident-l = equivToPath equi
+
+  module _ where
+    private
+      forwards : ((a , b) ∈ S)
+        → (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+      forwards proof = b , (proof , refl)
+
+      backwards : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+        → (a , b) ∈ S
+      backwards (b' , (ab'∈S , b'=b)) = subst b'=b ab'∈S
+
+      isbijective : (x : (a , b) ∈ S) → (backwards ∘ forwards) x ≡ x
+      isbijective x = pathJprop (λ y _ → y) x
+
+      fwd-bwd : (x : Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+        → (forwards ∘ backwards) x ≡ x
+      fwd-bwd (b , (ab∈S , refl)) = pathJprop (λ y _ → fst (snd y)) ab∈S
+
+      isequiv : isEquiv
+        (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+        ((a , b) ∈ S)
+        backwards
+      isequiv ab∈S = gradLemma backwards forwards isbijective fwd-bwd ab∈S
+
+      equi : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+        ≃ ab ∈ S
+      equi = backwards , isequiv
+
+    ident-r : (Σ[ b' ∈ B ] (a , b') ∈ S × (b' , b) ∈ Diag B)
+      ≡ ab ∈ S
+    ident-r = equivToPath equi
+
+Rel-as-Cat : Category
+Rel-as-Cat = record
+  { Object = Set
+  ; Arrow = λ S R → Subset (S × R)
+  ; 𝟙 = λ {S} → Diag S
+  ; _⊕_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
+  ; assoc = {!!}
+  ; ident = funExt ident-l , funExt ident-r
+  }
+
+module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ}} where
+  private
+    C-Obj = Object ℂ
+    _+_   = Arrow ℂ
+
+  RepFunctor : Functor ℂ 𝕊et-as-Cat
+  RepFunctor =
+    record
+      { F = λ A → (B : C-Obj) → Hom {ℂ = ℂ} A B
+      ; f = λ { {c' = c'} f g → HomFromArrow {ℂ = {!𝕊et-as-Cat!}} c' g}
+      ; ident = {!!}
+      ; distrib = {!!}
+      }
diff --git a/src/PathPrelude.agda b/src/PathPrelude.agda
deleted file mode 100644
index 0746fe1..0000000
--- a/src/PathPrelude.agda
+++ /dev/null
@@ -1,293 +0,0 @@
-{-# OPTIONS --cubical #-}
-module PathPrelude where
-
-  open import Primitives public
-  open import Level
-  open import Data.Product using (Σ; _,_) renaming (proj₁ to fst; proj₂ to snd)
-
-  refl : ∀ {a} {A : Set a} {x : A} → Path x x
-  refl {x = x} = λ i → x
-
-  sym : ∀ {a} {A : Set a} → {x y : A} → Path x y → Path y x
-  sym p = \ i → p (~ i)
-
-  pathJ : ∀ {a}{p}{A : Set a}{x : A}(P : ∀ y → Path x y → Set p) → P x ((\ i -> x)) → ∀ y (p : Path x y) → P y p
-  pathJ P d _ p = primComp (λ i → P (p i) (\ j → p (i ∧ j))) i0 (\ _ → empty) d
-
-
-  pathJprop : ∀ {a}{p}{A : Set a}{x : A}(P : ∀ y → Path x y → Set p) → (d : P x ((\ i -> x))) → pathJ P d _ refl ≡ d
-  pathJprop {x = x} P d i = primComp (λ _ → P x refl) i (\ { j (i = i1) → d }) d
-
-
-  trans : ∀ {a} {A : Set a} → {x y z : A} → Path x y → Path y z → Path x z
-  trans {A = A} {x} {y} p q = \ i → primComp (λ j → A) (i ∨ ~ i)
-                                             (\ j → \ { (i = i1) → q j
-                                                      ; (i = i0) → x
-                                                      }
-                                             )
-                                             (p i)
-
-  fun-ext : ∀ {a b} {A : Set a} {B : A → Set b} → {f g : (x : A) → B x}
-            → (∀ x → Path (f x) (g x)) → Path f g
-  fun-ext p = λ i x → p x i
-
-
-  -- comp using only Path
-  compP : ∀ {a : Level} {A0 A1 : Set a} (A : Path A0 A1) → {φ : I} → (a0 : A i0) → (Partial (Σ A1 \ y → PathP (\ i → A i) a0 y) φ) → A i1
-  compP A {φ} a0 p = primComp (λ i → A i) φ (\ i o → p o .snd i) a0
-
-  -- fill using only Path
-
-  fillP : ∀ {a : Level} {A0 A1 : Set a} (A : Path A0 A1) → {φ : I} → (a0 : A i0) → (Partial (Σ A1 \ y → PathP (\ i → A i) a0 y) φ) → ∀ i → A i
-  fillP A {φ} a0 p j = primComp (λ i → A (i ∧ j)) (φ ∨ ~ j) (\ { i (φ = i1) → p itIsOne .snd (i ∧ j); i (j = i0) → a0 }) a0
-
-
-  reflId : ∀ {a} {A : Set a}{x : A} → Id x x
-  reflId {x = x} = conid i1 (λ _ → x)
-
-  Jdef : ∀ {a}{p}{A : Set a}{x : A}(P : ∀ y → Id x y → Set p) → (d : P x reflId) → J P d reflId ≡ d
-  Jdef P d = refl
-
-  fromPath : ∀ {A : Set}{x y : A} → Path x y -> Id x y
-  fromPath p = conid i0 (\ i → p i)
-
-  transId : ∀ {a} {A : Set a} → {x y z : A} → Id x y → Id y z → Id x z
-  transId {A = A} {x} {y} p = J (λ y _ → Id x y) p
-
-  congId :  ∀ {a b} {A : Set a} {B : Set b} (f : A → B) → ∀ {x y} → Id x y → Id (f x) (f y)
-  congId f {x} {y} = J (λ y _ → Id (f x) (f y)) reflId
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-  fill : ∀ {a : I -> Level} (A : (i : I) → Set (a i)) (φ : I) → ((i : I) → Partial (A i) φ) → A i0 → (i : I) → A i
-  fill A φ u a0 i = unsafeComp (\ j → A (i ∧ j)) (φ ∨ ~ i) (\ j → unsafePOr φ (~ i) (u (i ∧ j)) \ { _ → a0 }) a0
-
-  singl : ∀ {l} {A : Set l} (a : A) -> Set l
-  singl {A = A} a = Σ A (\ x → a ≡ x)
-
-  contrSingl : ∀ {l} {A : Set l} {a b : A} (p : a ≡ b) → Path {A = singl a} (a , refl) (b , p)
-  contrSingl p = \ i → ((p i) , \ j →  p (i ∧ j))
-
-
-  module Contr where
-
-    isContr : ∀ {a} → (A : Set a) → Set a
-    isContr A = Σ A (\ x → ∀ y → x ≡ y)
-
-    contr : ∀ {a} {A : Set a} → isContr A → (φ : I) → (u : Partial A φ) → A
-    contr {A = A} (c , p) φ u = primComp (\ _ → A) φ (λ i → \ o → p (u o) i) c
-
-    lemma : ∀ {a} {A : Set a}
-            → (contr1 : ∀ φ → Partial A φ → A)
-            → (contr2 : ∀ u → u ≡ (contr1 i1 \{_ → u}))
-            → isContr A
-    lemma {A = A} contr1 contr2 = x , (λ y → let module M = R y in trans (contr2 x) (trans (\ i → M.v i) (sym (contr2 y)))) where
-          x = contr1 i0 empty
-          module R (y : A) (i : I) where
-            φ = ~ i ∨ i
-            u : Partial A φ
-            u = primPOr (~ i) i (\{_ → x}) (\{_ → y})
-            v = contr1 φ u
-
-    isContrProp : ∀ {a} {A : Set a} (h : isContr A) -> ∀ (x y : A) → x ≡ y
-    isContrProp {A = A} h a b = \ i → primComp (\ _ → A) _ (\ j → [ (~ i) ↦ (\{_ → snd h a j}) , i ↦ (\{_ → snd h b j}) ]) (fst h)
-
-  module Pres {la lb : I → _} {T : (i : I) → Set (lb i)}{A : (i : I) → Set (la i)} (f : ∀ i → T i → A i) (φ : I) (t : ∀ i → Partial (T i) φ)
-                  (t0 : T i0 {- [ φ ↦ t i0 ] -}) where
-
-   c1 c2 : A i1
-   c1 = unsafeComp A φ (λ i → (\{_ → f i (t i itIsOne) })) (f i0 t0)
-   c2 = f i1 (unsafeComp T φ t t0)
-
-   a0 = f i0 t0
-
-   a : ∀ i → Partial (A i) φ
-   a i = (\{_ → f i ((t i) itIsOne) })
-
-   u : ∀ i → A i
-   u = fill A φ a a0
-
-   v : ∀ i → T i
-   v = fill T φ t t0
-
-   pres : Path c1 c2
-   pres = \ j → unsafeComp A (φ ∨ (j ∨ ~ j)) (λ i → primPOr φ ((j ∨ ~ j)) (a i) (primPOr j (~ j)  (\{_ →  f i (v i)  }) (\{_ →  u i  }))) a0
-
-  module Equiv {l l'} (A : Set l)(B : Set l') where
-    isEquiv : (A -> B) → Set (l' ⊔ l)
-    isEquiv f = ∀ y → Contr.isContr (Σ A \ x → y ≡ f x)
-
-    Equiv = Σ _ isEquiv
-
-    equiv : (f : Equiv) → ∀ φ (t : Partial A φ) (a : B {- [ φ ↦ f t ] -}) → PartialP φ (\ o → Path a (fst f (t o)))
-             -> Σ A \ x → a ≡ fst f x -- [ φ ↦ (t , \ j → a) ]
-    equiv (f , [f]) φ t a p = Contr.contr ([f] a) φ \ o → t o , (\ i → p o i)
-
-    equiv1 : (f : Equiv) → ∀ φ (t : Partial A φ) (a : B {- [ φ ↦ f t ] -}) → PartialP φ (\ o → Path a (fst f (t o))) -> A
-    equiv1 f φ t a p = fst (equiv f φ t a p)
-
-    equiv2 : (f : Equiv) → ∀ φ (t : Partial A φ) (a : B {- [ φ ↦ f t ] -}) → (p : PartialP φ (\ o → Path a (fst f (t o))))
-             → a ≡ fst f (equiv1 f φ t a p)
-    equiv2 f φ t a p = snd (equiv f φ t a p)
-
-  open Equiv public
-
-  {-# BUILTIN ISEQUIV isEquiv #-}
-
-  idEquiv : ∀ {a} {A : Set a} → Equiv A A
-  idEquiv = (λ x → x) , (λ y → (y , refl) , (λ y₁ → contrSingl (snd y₁)))
-
-
-  pathToEquiv : ∀ {l : I → _} (E : (i : I) → Set (l i)) → Equiv (E i0) (E i1)
-  pathToEquiv E = f
-                , (λ y → (g y
-                         , (\ j → primComp E (~ j ∨ j) (\ i → [ ~ j ↦  (\{_ → v i y })  , j ↦  (\{_ → u i (g y) })  ]) (g y))) ,
-                                          prop-f-image y _ )
-    where
-      A = E i0
-      B = E i1
-      transp : ∀ {l : I → _} (E : (i : I) → Set (l i)) → E i0 → E i1
-      transp E x = primComp E i0 (\ _ → empty) x
-      f : A → B
-      f = transp E
-      g : B → A
-      g = transp (\ i → E (~ i))
-      u : (i : I) → A → E i
-      u i x = fill E i0 (\ _ → empty) x i
-      v : (i : I) → B → E i
-      v i y = fill (\ i → E (~ i)) i0 (\ _ → empty) y (~ i)
-      prop-f-image : (y : B) → (a b : Σ _ \ x → y ≡ f x) → a ≡ b
-      prop-f-image y (x0 , b0) (x1 , b1) = \ k → (w k) , (\ j → d j k)
-         where
-           w0 = \ (j : I) → primComp (\ i → E (~ i)) (~ j ∨ j) ((\ i → [ ~ j ↦  (\{_ → v (~ i) y })  , j ↦  (\{_ → u (~ i) x0 })  ])) (b0 j)
-           w1 = \ (j : I) → primComp (\ i → E (~ i)) (~ j ∨ j) ((\ i → [ ~ j ↦  (\{_ → v (~ i) y })  , j ↦  (\{_ → u (~ i) x1 })  ])) (b1 j)
-           t0 = \ (j : I) → fill (\ i → E (~ i)) (~ j ∨ j) ((\ i → [ ~ j ↦  (\{_ → v (~ i) y })  , j ↦  (\{_ → u (~ i) x0 })  ])) (b0 j)
-           t1 = \ (j : I) → fill (\ i → E (~ i)) (~ j ∨ j) ((\ i → [ ~ j ↦  (\{_ → v (~ i) y })  , j ↦  (\{_ → u (~ i) x1 })  ])) (b1 j)
-           w = \ (k : I) → primComp (λ _ → A) (~ k ∨ k) (\ j → [ ~ k ↦ (\{_ → w0 j }) , k ↦ (\{_ → w1 j }) ]) (g y)
-           t = \ (j k : I) → fill (λ _ → A) (~ k ∨ k) (\ j → [ ~ k ↦ (\{_ → w0 j }) , k ↦ (\{_ → w1 j }) ]) (g y) j
-           d = \ (j k : I) → primComp E ((~ k ∨ k) ∨ (~ j ∨ j)) ((\ i → [ ~ k ∨ k ↦ [ ~ k ↦  (\{_ →  t0 j (~ i)  })  , k ↦  (\{_ → t1 j (~ i) })  ]
-                                                     , ~ j ∨ j ↦ [ ~ j ↦  (\{_ → v (i) y })  , j ↦  (\{_ → u (i) (w k) })  ] ])) (t j k)
-
-  pathToEquiv2 : ∀ {l : I → _} (E : (i : I) → Set (l i)) → _
-  pathToEquiv2 {l} E = snd (pathToEquiv E)
-
-  {-# BUILTIN PATHTOEQUIV pathToEquiv2 #-}
-
-  module GluePrims where
-   primitive
-    primGlue : ∀ {a b} (A : Set a) → ∀ φ → (T : Partial (Set b) φ) → (f : PartialP φ (λ o → T o → A))
-                                           → (pf : PartialP φ (λ o → isEquiv (T o) A (f o))) → Set b
-    prim^glue : ∀ {a b} {A : Set a} {φ : I} {T : Partial (Set b) φ}
-                  {f : PartialP φ (λ o → T o → A)}
-                  {pf : PartialP φ (λ o → isEquiv (T o) A (f o))} →
-                  PartialP φ T → A → primGlue A φ T f pf
-    prim^unglue : ∀ {a b} {A : Set a} {φ : I} {T : Partial (Set b) φ}
-                    {f : PartialP φ (λ o → T o → A)}
-                    {pf : PartialP φ (λ o → isEquiv (T o) A (f o))} →
-                    primGlue A φ T f pf → A
-
-
-  module GluePrims' (dummy : Set₁) = GluePrims
-  open GluePrims' Set using () renaming (prim^glue to unsafeGlue) public
-
-  open GluePrims public renaming (prim^glue to glue; prim^unglue to unglue)
-
-  Glue : ∀ {a b} (A : Set a) → ∀ φ → (T : Partial (Set b) φ) (f : (PartialP φ \ o → Equiv (T o) A))  → Set _
-  Glue A φ T f = primGlue A φ T (\ o → fst (f o)) (\ o → snd (f o))
-
-  eqToPath' : ∀ {l} {A B : Set l} → Equiv A B → Path A B
-  eqToPath' {l} {A} {B} f = \ i → Glue B (~ i ∨ i) ([ ~ i ↦ (\ _ → A) , i ↦ (\ _ → B) ]) ([ ~ i ↦ (\{_ → f }) , i ↦ (\{_ → idEquiv }) ])
-
-  primitive
-    primFaceForall : (I → I) → I
-
-  module FR (φ : I -> I) where
-     δ = primFaceForall φ
-     postulate
-         ∀e : IsOne δ → ∀ i → IsOne (φ i)
-         ∀∨ : ∀ i → IsOne (φ i) → IsOne ((δ ∨ (φ i0 ∧ ~ i)) ∨ (φ i1 ∧ i))
-
-  module GlueIso {a b} {A : Set a} {φ : I} {T : Partial (Set b) φ} {f : (PartialP φ \ o → Equiv (T o) A)} where
-    going : PartialP φ (\ o → Glue A φ T f → T o)
-    going = (\{_ → (\ x → x) })
-    back : PartialP φ (\ o → T o → Glue A φ T f)
-    back = (\{_ → (\ x → x) })
-
-    lemma : ∀ (b : PartialP φ (\ _ → Glue A φ T f)) → PartialP φ \ o → Path (unglue {φ = φ} (b o)) (fst (f o) (going o (b o)))
-    lemma b = (\{_ → refl })
-
-  module CompGlue {la lb : I → _} (A : (i : I) → Set (la i)) (φ : I -> I) (T : ∀ i → Partial (Set (lb i)) (φ i))
-                           (f : ∀ i → PartialP (φ i) \ o → Equiv (T i o) (A i)) where
-     B : (i : I) -> Set (lb i)
-     B = \ i → Glue (A i) (φ i) (T i) (f i)
-     module Local (ψ : I) (b : ∀ i → Partial (B i) ψ) (b0 : B i0 {- [ ψ ↦ b i0 ] -}) where
-       a : ∀ i → Partial (A i) ψ
-       a i = \ o → unglue {φ = φ i} (b i o)
-       module Forall (δ : I) (∀e : IsOne δ → ∀ i → IsOne (φ i)) where
-         a0 : A i0
-         a0 = unglue {φ = φ i0} b0
-         a₁' = unsafeComp A ψ a a0
-         t₁' : PartialP δ (\ o → T i1 (∀e o i1))
-         t₁' = \ o → unsafeComp (λ i → T i (∀e o i)) ψ (\ i o' → GlueIso.going {φ = φ i} (∀e o i) (b i o')) (GlueIso.going {φ = φ i0} (∀e o i0) b0)
-         w : PartialP δ _
-         w = \ o → Pres.pres (\ i → fst (f i (∀e o i))) ψ (λ i x → GlueIso.going {φ = φ i} (∀e o i) (b i x)) (GlueIso.going {φ = φ i0} (∀e o i0) b0)
-         a₁'' = unsafeComp (\ _ → A i1) (δ ∨ ψ) (\ j → unsafePOr δ ψ (\ o → w o j) (a i1)) a₁'
-         g : PartialP (φ i1) _
-         g o = (equiv (T i1 _) (A i1) (f i1 o) (δ ∨ ψ) (unsafePOr δ ψ t₁' (\ o' → GlueIso.going {φ = φ i1} o (b i1 o'))) a₁''
-                          ( (unsafePOr δ ψ (\{ (δ = i1) → refl })  ((\{ (ψ = i1) → GlueIso.lemma {φ = φ i1} (\ _ → b i1 itIsOne) o })  ) ) ))
-                          -- TODO figure out why we need (δ = i1) and (ψ = i1) here
-         t₁ : PartialP (φ i1) _
-         t₁ o = fst (g o)
-         α : PartialP (φ i1) _
-         α o = snd (g o)
-         a₁ = unsafeComp (\ j → A i1) (φ i1 ∨ ψ) (\ j → unsafePOr (φ i1) ψ (\ o → α o j) (a i1)) a₁''
-         b₁ : Glue _ (φ i1) (T i1) (f i1)
-         b₁ = unsafeGlue {φ = φ i1} t₁ a₁
-       b1 = Forall.b₁ (FR.δ φ) (FR.∀e φ)
-
-  compGlue : {la lb : I → _} (A : (i : I) → Set (la i)) (φ : I -> I) (T : ∀ i → Partial (Set (lb i)) (φ i))
-                           (f : ∀ i → PartialP (φ i) \ o → (T i o) → (A i)) →
-                           (pf : ∀ i → PartialP (φ i) (λ o → isEquiv (T i o) (A i) (f i o))) →
-             let
-                B : (i : I) -> Set (lb i)
-                B = \ i → primGlue (A i) (φ i) (T i) (f i) (pf i)
-             in  (ψ : I) (b : ∀ i → Partial (B i) ψ) (b0 : B i0) → B i1
-  compGlue A φ T f pf ψ b b0 = CompGlue.Local.b1 A φ T (λ i p → (f i p) , (pf i p)) ψ b b0
-
-
-  {-# BUILTIN COMPGLUE compGlue #-}
-
-  module ≡-Reasoning {a} {A : Set a} where
-
-    infix  3 _∎
-    infixr 2 _≡⟨⟩_ _≡⟨_⟩_ -- _≅⟨_⟩_
-    infix  1 begin_
-
-    begin_ : ∀{x y : A} → x ≡ y → x ≡ y
-    begin_ x≡y = x≡y
-
-    _≡⟨⟩_ : ∀ (x {y} : A) → x ≡ y → x ≡ y
-    _ ≡⟨⟩ x≡y = x≡y
-
-    _≡⟨_⟩_ : ∀ (x {y z} : A) → x ≡ y → y ≡ z → x ≡ z
-    _ ≡⟨ x≡y ⟩ y≡z = trans x≡y y≡z
-
-    -- _≅⟨_⟩_ : ∀ (x {y z} : A) → x ≅ y → y ≡ z → x ≡ z
-    -- _ ≅⟨ x≅y ⟩ y≡z = trans (H.≅-to-≡ x≅y) y≡z
-
-    _∎ : ∀ (x : A) → x ≡ x
-    _∎ _ = refl
diff --git a/src/Primitives.agda b/src/Primitives.agda
deleted file mode 100644
index 258b1f7..0000000
--- a/src/Primitives.agda
+++ /dev/null
@@ -1,62 +0,0 @@
-{-# OPTIONS --cubical #-}
-module Primitives where
-
-
-
-module Postulates where
-  open import Agda.Primitive public
-  open CubicalPrimitives public renaming (isOneEmpty to empty)
-
-  postulate
-    Path : ∀ {a} {A : Set a} → A → A → Set a
-    PathP : ∀ {a} → (A : I → Set a) → A i0 → A i1 → Set a
-
-  {-# BUILTIN PATH         Path     #-}
-  {-# BUILTIN PATHP        PathP     #-}
-
-  primitive
-    primPathApply : ∀ {a} {A : Set a} {x y : A} → Path x y → I → A
-    primPathPApply : ∀ {a} → {A : I → Set a} → ∀ {x y} → PathP A x y → (i : I) → A i
-
-  postulate
-    Id : ∀ {a} {A : Set a} → A → A → Set a
-
-  {-# BUILTIN ID           Id       #-}
-  {-# BUILTIN CONID        conid    #-}
-
-  primitive
-    primDepIMin : _
-    primIdFace : {a : Level} {A : Set a} {x y : A} → Id x y → I
-    primIdPath : {a : Level} {A : Set a} {x y : A} → Id x y → Path x y
-
-  primitive
-    primIdJ : ∀ {a}{p}{A : Set a}{x : A}(P : ∀ y → Id x y → Set p) → P x (conid i1 (\ i -> x)) → ∀ {y} (p : Id x y) → P y p
-
-  {-# BUILTIN SUB Sub #-}
-
-  postulate
-    inc : ∀ {a} {A : Set a} {φ} (x : A) → Sub {A = A} φ (\ _ → x)
-
-  {-# BUILTIN SUBIN inc #-}
-
-  primitive
-    primSubOut : {a : Level} {A : Set a} {φ : I} {u : Partial A φ} → Sub φ u → A
-
-
-open Postulates public renaming (primPathApply to _∙_; primIMin to _∧_; primIMax to _∨_; primINeg to ~_
-                                ; primPFrom1 to p[_]
-                                ; primIdJ to J
-                                ; primSubOut to ouc
-                                ; Path to Path'
-                                )
-
-module Unsafe' (dummy : Set₁) = Postulates
-
-unsafeComp = Unsafe'.primComp Set
-unsafePOr = Unsafe'.primPOr Set
-
-Path : ∀ {a} {A : Set a} → A → A → Set a
-Path {A = A} = PathP (\ _ → A)
-
-infix 4 _≡_
-_≡_ = Path