Rename \o to <<<

src/Cat/Categories/Cat.agda

@@ -18,7 +18,7 @@ module _ (ℓ ℓ' : Level) where
   RawCategory.Object   RawCat = Category ℓ ℓ'
   RawCategory.Arrow    RawCat = Functor
   RawCategory.identity RawCat = Functors.identity
-  RawCategory._∘_      RawCat = F[_∘_]
+  RawCategory._<<<_    RawCat = F[_∘_]
 
   -- NB! `ArrowsAreSets RawCat` is *not* provable. The type of functors,
   -- however, form a groupoid! Therefore there is no (1-)category of
@@ -50,18 +50,18 @@ module CatProduct {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
         Arr (c , d) (c' , d') = ℂ [ c , c' ] × 𝔻 [ d , d' ]
         identity : {o : Obj} → Arr o o
         identity = ℂ.identity , 𝔻.identity
-        _∘_ :
+        _<<<_ :
           {a b c : Obj} →
           Arr b c →
           Arr a b →
           Arr a c
-        _∘_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → ℂ [ bc∈C ∘ ab∈C ] , 𝔻 [ bc∈D ∘ ab∈D ]}
+        _<<<_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → ℂ [ bc∈C ∘ ab∈C ] , 𝔻 [ bc∈D ∘ ab∈D ]}
 
       rawProduct : RawCategory ℓ ℓ'
       RawCategory.Object   rawProduct = Obj
       RawCategory.Arrow    rawProduct = Arr
       RawCategory.identity rawProduct = identity
-      RawCategory._∘_      rawProduct = _∘_
+      RawCategory._<<<_    rawProduct = _<<<_
 
     open RawCategory rawProduct
 

src/Cat/Categories/Cube.agda

@@ -19,7 +19,7 @@ open import Cat.Category.Functor
 -- See section 6.8 in Huber's thesis for details on how to implement the
 -- categorical version of CTT
 
-open Category hiding (_∘_)
+open Category hiding (_<<<_)
 open Functor
 
 module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
@@ -68,7 +68,7 @@ module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
     Raw.Object Rawℂ = FiniteDecidableSubset
     Raw.Arrow Rawℂ = Hom
     Raw.identity Rawℂ {o} = inj₁ , λ { (i , ii) (j , jj) eq → Σ≡ eq {!refl!} }
-    Raw._∘_ Rawℂ = {!!}
+    Raw._<<<_ Rawℂ = {!!}
 
     postulate IsCategoryℂ : IsCategory Rawℂ
 

src/Cat/Categories/Fam.agda

@@ -14,15 +14,15 @@ module _ (ℓa ℓb : Level) where
     identity : {A : Object} → Arr A A
     fst identity = λ x → x
     snd identity = λ b → b
-    _∘_ : {a b c : Object} → Arr b c → Arr a b → Arr a c
+    _<<<_ : {a b c : Object} → Arr b c → Arr a b → Arr a c
-    (g , g') ∘ (f , f') = g Function.∘ f , g' Function.∘ f'
+    (g , g') <<< (f , f') = g Function.∘ f , g' Function.∘ f'
 
     RawFam : RawCategory (lsuc (ℓa ⊔ ℓb)) (ℓa ⊔ ℓb)
     RawFam = record
       { Object = Object
       ; Arrow = Arr
       ; identity = λ { {A} → identity {A = A}}
-      ; _∘_ = λ {a b c} → _∘_ {a} {b} {c}
+      ; _<<<_ = λ {a b c} → _<<<_ {a} {b} {c}
       }
 
     open RawCategory RawFam hiding (Object ; identity)

src/Cat/Categories/Free.agda

@@ -30,7 +30,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
     RawCategory.Object   RawFree = ℂ.Object
     RawCategory.Arrow    RawFree = Path ℂ.Arrow
     RawCategory.identity RawFree = empty
-    RawCategory._∘_      RawFree = concatenate
+    RawCategory._<<<_    RawFree = concatenate
 
     open RawCategory RawFree
 

src/Cat/Categories/Fun.agda

@@ -20,7 +20,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
       RawCategory.Object   raw = Functor ℂ 𝔻
       RawCategory.Arrow    raw = NaturalTransformation
       RawCategory.identity raw {F} = identity F
-      RawCategory._∘_      raw {F} {G} {H} = NT[_∘_] {F} {G} {H}
+      RawCategory._<<<_    raw {F} {G} {H} = NT[_∘_] {F} {G} {H}
 
     module _ where
       open RawCategory raw hiding (identity)
@@ -154,9 +154,9 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
         ob = fromEq p
         re : Arrow B A
         re = fromEq (sym p)
-        vr : _∘_ {A = A} {B} {A} re ob ≡ identity A
+        vr : _<<<_ {A = A} {B} {A} re ob ≡ identity A
         vr = {!!}
-        rv : _∘_ {A = B} {A} {B} ob re ≡ identity B
+        rv : _<<<_ {A = B} {A} {B} ob re ≡ identity B
         rv = {!!}
         isInverse : IsInverseOf {A} {B} ob re
         isInverse = vr , rv
@@ -201,7 +201,7 @@ module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
       { Object = Presheaf ℂ
       ; Arrow = NaturalTransformation
       ; identity = λ {F} → identity F
-      ; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
+      ; _<<<_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
       }
 
   --   isCategory : IsCategory raw

src/Cat/Categories/Rel.agda

@@ -154,7 +154,7 @@ RawRel = record
   { Object = Set
   ; Arrow = λ S R → Subset (S × R)
   ; identity = λ {S} → Diag S
-  ; _∘_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
+  ; _<<<_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
   }
 
 isPreCategory : IsPreCategory RawRel

src/Cat/Categories/Sets.agda

@@ -26,11 +26,11 @@ module _ (ℓ : Level) where
     RawCategory.Object   SetsRaw = hSet ℓ
     RawCategory.Arrow    SetsRaw (T , _) (U , _) = T → U
     RawCategory.identity SetsRaw = Function.id
-    RawCategory._∘_      SetsRaw = Function._∘′_
+    RawCategory._<<<_    SetsRaw = Function._∘′_
 
     module _ where
       private
-        open RawCategory SetsRaw hiding (_∘_)
+        open RawCategory SetsRaw hiding (_<<<_)
 
         isIdentity : IsIdentity Function.id
         fst isIdentity = funExt λ _ → refl
@@ -44,7 +44,7 @@ module _ (ℓ : Level) where
       IsPreCategory.isIdentity    isPreCat {A} {B} = isIdentity    {A} {B}
       IsPreCategory.arrowsAreSets isPreCat {A} {B} = arrowsAreSets {A} {B}
 
-    open IsPreCategory isPreCat hiding (_∘_)
+    open IsPreCategory isPreCat hiding (_<<<_)
 
     isIso = TypeIsomorphism
     module _ {hA hB : hSet ℓ} where

src/Cat/Category.agda

@@ -13,7 +13,7 @@
 -- Data
 -- ----
 -- identity; the identity arrow
--- _∘_; function composition
+-- _<<<_; function composition
 --
 -- Laws
 -- ----
@@ -52,9 +52,9 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
     Object   : Set ℓa
     Arrow    : Object → Object → Set ℓb
     identity : {A : Object} → Arrow A A
-    _∘_      : {A B C : Object} → Arrow B C → Arrow A B → Arrow A C
+    _<<<_    : {A B C : Object} → Arrow B C → Arrow A B → Arrow A C
 
-  infixl 10 _∘_ _>>>_
+  infixl 10 _<<<_ _>>>_
 
   -- | Operations on data
 
@@ -65,7 +65,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
   codomain {b = b} _ = b
 
   _>>>_ : {A B C : Object} → (Arrow A B) → (Arrow B C) → Arrow A C
-  f >>> g = g ∘ f
+  f >>> g = g <<< f
 
   -- | Laws about the data
 
@@ -73,17 +73,17 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
   -- right-hand-side.
   IsAssociative : Set (ℓa ⊔ ℓb)
   IsAssociative = ∀ {A B C D} {f : Arrow A B} {g : Arrow B C} {h : Arrow C D}
-    → h ∘ (g ∘ f) ≡ (h ∘ g) ∘ f
+    → h <<< (g <<< f) ≡ (h <<< g) <<< f
 
   IsIdentity : ({A : Object} → Arrow A A) → Set (ℓa ⊔ ℓb)
   IsIdentity id = {A B : Object} {f : Arrow A B}
-    → id ∘ f ≡ f × f ∘ id ≡ f
+    → id <<< f ≡ f × f <<< id ≡ f
 
   ArrowsAreSets : Set (ℓa ⊔ ℓb)
   ArrowsAreSets = ∀ {A B : Object} → isSet (Arrow A B)
 
   IsInverseOf : ∀ {A B} → (Arrow A B) → (Arrow B A) → Set ℓb
-  IsInverseOf = λ f g → g ∘ f ≡ identity × f ∘ g ≡ identity
+  IsInverseOf = λ f g → g <<< f ≡ identity × f <<< g ≡ identity
 
   Isomorphism : ∀ {A B} → (f : Arrow A B) → Set ℓb
   Isomorphism {A} {B} f = Σ[ g ∈ Arrow B A ] IsInverseOf f g
@@ -93,10 +93,10 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
 
   module _ {A B : Object} where
     Epimorphism : {X : Object } → (f : Arrow A B) → Set ℓb
-    Epimorphism {X} f = (g₀ g₁ : Arrow B X) → g₀ ∘ f ≡ g₁ ∘ f → g₀ ≡ g₁
+    Epimorphism {X} f = (g₀ g₁ : Arrow B X) → g₀ <<< f ≡ g₁ <<< f → g₀ ≡ g₁
 
     Monomorphism : {X : Object} → (f : Arrow A B) → Set ℓb
-    Monomorphism {X} f = (g₀ g₁ : Arrow X A) → f ∘ g₀ ≡ f ∘ g₁ → g₀ ≡ g₁
+    Monomorphism {X} f = (g₀ g₁ : Arrow X A) → f <<< g₀ ≡ f <<< g₁ → g₀ ≡ g₁
 
   IsInitial  : Object → Set (ℓa ⊔ ℓb)
   IsInitial  I = {X : Object} → isContr (Arrow I X)
@@ -167,10 +167,10 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
       arrowsAreSets : ArrowsAreSets
     open Univalence isIdentity public
 
-    leftIdentity : {A B : Object} {f : Arrow A B} → identity ∘ f ≡ f
+    leftIdentity : {A B : Object} {f : Arrow A B} → identity <<< f ≡ f
     leftIdentity {A} {B} {f} = fst (isIdentity {A = A} {B} {f})
 
-    rightIdentity : {A B : Object} {f : Arrow A B} → f ∘ identity ≡ f
+    rightIdentity : {A B : Object} {f : Arrow A B} → f <<< identity ≡ f
     rightIdentity {A} {B} {f} = snd (isIdentity {A = A} {B} {f})
 
     ------------
@@ -181,26 +181,26 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
     module _ {A B : Object} {X : Object} (f : Arrow A B) where
       iso→epi : Isomorphism f → Epimorphism {X = X} f
       iso→epi (f- , left-inv , right-inv) g₀ g₁ eq = begin
-        g₀              ≡⟨ sym rightIdentity ⟩
+        g₀                  ≡⟨ sym rightIdentity ⟩
-        g₀ ∘ identity   ≡⟨ cong (_∘_ g₀) (sym right-inv) ⟩
+        g₀ <<< identity     ≡⟨ cong (_<<<_ g₀) (sym right-inv) ⟩
-        g₀ ∘ (f ∘ f-)   ≡⟨ isAssociative ⟩
+        g₀ <<< (f <<< f-)   ≡⟨ isAssociative ⟩
-        (g₀ ∘ f) ∘ f-   ≡⟨ cong (λ φ → φ ∘ f-) eq ⟩
+        (g₀ <<< f) <<< f-   ≡⟨ cong (λ φ → φ <<< f-) eq ⟩
-        (g₁ ∘ f) ∘ f-   ≡⟨ sym isAssociative ⟩
+        (g₁ <<< f) <<< f-   ≡⟨ sym isAssociative ⟩
-        g₁ ∘ (f ∘ f-)   ≡⟨ cong (_∘_ g₁) right-inv ⟩
+        g₁ <<< (f <<< f-)   ≡⟨ cong (_<<<_ g₁) right-inv ⟩
-        g₁ ∘ identity   ≡⟨ rightIdentity ⟩
+        g₁ <<< identity     ≡⟨ rightIdentity ⟩
-        g₁              ∎
+        g₁                  ∎
 
       iso→mono : Isomorphism f → Monomorphism {X = X} f
       iso→mono (f- , left-inv , right-inv) g₀ g₁ eq =
         begin
-        g₀            ≡⟨ sym leftIdentity ⟩
+        g₀                ≡⟨ sym leftIdentity ⟩
-        identity ∘ g₀ ≡⟨ cong (λ φ → φ ∘ g₀) (sym left-inv) ⟩
+        identity <<< g₀   ≡⟨ cong (λ φ → φ <<< g₀) (sym left-inv) ⟩
-        (f- ∘ f) ∘ g₀ ≡⟨ sym isAssociative ⟩
+        (f- <<< f) <<< g₀ ≡⟨ sym isAssociative ⟩
-        f- ∘ (f ∘ g₀) ≡⟨ cong (_∘_ f-) eq ⟩
+        f- <<< (f <<< g₀) ≡⟨ cong (_<<<_ f-) eq ⟩
-        f- ∘ (f ∘ g₁) ≡⟨ isAssociative ⟩
+        f- <<< (f <<< g₁) ≡⟨ isAssociative ⟩
-        (f- ∘ f) ∘ g₁ ≡⟨ cong (λ φ → φ ∘ g₁) left-inv ⟩
+        (f- <<< f) <<< g₁ ≡⟨ cong (λ φ → φ <<< g₁) left-inv ⟩
-        identity ∘ g₁ ≡⟨ leftIdentity ⟩
+        identity <<< g₁   ≡⟨ leftIdentity ⟩
-        g₁            ∎
+        g₁                ∎
 
       iso→epi×mono : Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
       iso→epi×mono iso = iso→epi iso , iso→mono iso
@@ -210,7 +210,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
 
     propIsIdentity : ∀ {f : ∀ {A} → Arrow A A} → isProp (IsIdentity f)
     propIsIdentity {id} = propPiImpl (λ _ → propPiImpl λ _ → propPiImpl (λ f →
-      propSig (arrowsAreSets (id ∘ f) f) λ _ → arrowsAreSets (f ∘ id) f))
+      propSig (arrowsAreSets (id <<< f) f) λ _ → arrowsAreSets (f <<< id) f))
 
     propArrowIsSet : isProp (∀ {A B} → isSet (Arrow A B))
     propArrowIsSet = propPiImpl λ _ → propPiImpl (λ _ → isSetIsProp)
@@ -225,12 +225,12 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
           where
             geq : g ≡ g'
             geq = begin
-              g             ≡⟨ sym rightIdentity ⟩
+              g                 ≡⟨ sym rightIdentity ⟩
-              g ∘ identity  ≡⟨ cong (λ φ → g ∘ φ) (sym ε') ⟩
+              g <<< identity    ≡⟨ cong (λ φ → g <<< φ) (sym ε') ⟩
-              g ∘ (f ∘ g')  ≡⟨ isAssociative ⟩
+              g <<< (f <<< g')  ≡⟨ isAssociative ⟩
-              (g ∘ f) ∘ g'  ≡⟨ cong (λ φ → φ ∘ g') η ⟩
+              (g <<< f) <<< g'  ≡⟨ cong (λ φ → φ <<< g') η ⟩
-              identity ∘ g' ≡⟨ leftIdentity ⟩
+              identity <<< g'   ≡⟨ leftIdentity ⟩
-              g'            ∎
+              g'                ∎
 
     propIsInitial : ∀ I → isProp (IsInitial I)
     propIsInitial I x y i {X} = res X i
@@ -266,23 +266,23 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
       private
         trans≅ : Transitive _≅_
         trans≅ (f , f~ , f-inv) (g , g~ , g-inv)
-          = g ∘ f
+          = g <<< f
-          , f~ ∘ g~
+          , f~ <<< g~
           , ( begin
-              (f~ ∘ g~) ∘ (g ∘ f) ≡⟨ isAssociative ⟩
+              (f~ <<< g~) <<< (g <<< f) ≡⟨ isAssociative ⟩
-              (f~ ∘ g~) ∘ g ∘ f ≡⟨ cong (λ φ → φ ∘ f) (sym isAssociative) ⟩
+              (f~ <<< g~) <<< g <<< f   ≡⟨ cong (λ φ → φ <<< f) (sym isAssociative) ⟩
-              f~ ∘ (g~ ∘ g) ∘ f ≡⟨ cong (λ φ → f~ ∘ φ ∘ f) (fst g-inv) ⟩
+              f~ <<< (g~ <<< g) <<< f   ≡⟨ cong (λ φ → f~ <<< φ <<< f) (fst g-inv) ⟩
-              f~ ∘ identity ∘ f ≡⟨ cong (λ φ → φ ∘ f) rightIdentity ⟩
+              f~ <<< identity <<< f     ≡⟨ cong (λ φ → φ <<< f) rightIdentity ⟩
-              f~ ∘ f           ≡⟨ fst f-inv ⟩
+              f~ <<< f                  ≡⟨ fst f-inv ⟩
-              identity ∎
+              identity                  ∎
             )
           , ( begin
-              g ∘ f ∘ (f~ ∘ g~) ≡⟨ isAssociative ⟩
+              g <<< f <<< (f~ <<< g~) ≡⟨ isAssociative ⟩
-              g ∘ f ∘ f~ ∘ g~ ≡⟨ cong (λ φ → φ ∘ g~) (sym isAssociative) ⟩
+              g <<< f <<< f~ <<< g~   ≡⟨ cong (λ φ → φ <<< g~) (sym isAssociative) ⟩
-              g ∘ (f ∘ f~) ∘ g~ ≡⟨ cong (λ φ → g ∘ φ ∘ g~) (snd f-inv) ⟩
+              g <<< (f <<< f~) <<< g~ ≡⟨ cong (λ φ → g <<< φ <<< g~) (snd f-inv) ⟩
-              g ∘ identity ∘ g~ ≡⟨ cong (λ φ → φ ∘ g~) rightIdentity ⟩
+              g <<< identity <<< g~   ≡⟨ cong (λ φ → φ <<< g~) rightIdentity ⟩
-              g ∘ g~ ≡⟨ snd g-inv ⟩
+              g <<< g~                ≡⟨ snd g-inv ⟩
-              identity ∎
+              identity                ∎
             )
         isPreorder : IsPreorder _≅_
         isPreorder = record { isEquivalence = equalityIsEquivalence ; reflexive = idToIso _ _ ; trans = trans≅ }
@@ -341,7 +341,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
         pq : Arrow a b ≡ Arrow a' b'
         pq i = Arrow (p i) (q i)
 
-      9-1-9 : coe pq f ≡ q* ∘ f ∘ p~
+      9-1-9 : coe pq f ≡ q* <<< f <<< p~
       9-1-9 = transpP {!!} {!!}
 
     -- | All projections are propositions.
@@ -366,9 +366,9 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
         Xprop f g = trans (sym (snd Xit f)) (snd Xit g)
         Yprop : isProp (Arrow Y Y)
         Yprop f g = trans (sym (snd Yit f)) (snd Yit g)
-        left : Y→X ∘ X→Y ≡ identity
+        left : Y→X <<< X→Y ≡ identity
         left = Xprop _ _
-        right : X→Y ∘ Y→X ≡ identity
+        right : X→Y <<< Y→X ≡ identity
         right = Yprop _ _
         iso : X ≅ Y
         iso = X→Y , Y→X , left , right
@@ -396,9 +396,9 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
         Xprop f g = trans (sym (snd Xii f)) (snd Xii g)
         Yprop : isProp (Arrow Y Y)
         Yprop f g = trans (sym (snd Yii f)) (snd Yii g)
-        left : Y→X ∘ X→Y ≡ identity
+        left : Y→X <<< X→Y ≡ identity
         left = Yprop _ _
-        right : X→Y ∘ Y→X ≡ identity
+        right : X→Y <<< Y→X ≡ identity
         right = Xprop _ _
         iso : X ≅ Y
         iso = Y→X , X→Y , right , left
@@ -497,7 +497,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
   _[_,_] = Arrow
 
   _[_∘_] : {A B C : Object} → (g : Arrow B C) → (f : Arrow A B) → Arrow A C
-  _[_∘_] = _∘_
+  _[_∘_] = _<<<_
 
 -- | The opposite category
 --
@@ -512,7 +512,7 @@ module Opposite {ℓa ℓb : Level} where
         RawCategory.Object   opRaw = ℂ.Object
         RawCategory.Arrow    opRaw = Function.flip ℂ.Arrow
         RawCategory.identity opRaw = ℂ.identity
-        RawCategory._∘_      opRaw = ℂ._>>>_
+        RawCategory._<<<_    opRaw = ℂ._>>>_
 
         open RawCategory opRaw
 
@@ -561,7 +561,7 @@ module Opposite {ℓa ℓb : Level} where
 
         -- inv : AreInverses (ℂ.idToIso A B) f
         inv-ζ : AreInverses (idToIso A B) ζ
-        -- recto-verso : ℂ.idToIso A B ∘ f ≡ idFun (A ℂ.≅ B)
+        -- recto-verso : ℂ.idToIso A B <<< f ≡ idFun (A ℂ.≅ B)
         inv-ζ = record
           { verso-recto = funExt (λ x → begin
             (ζ ⊙ idToIso A B) x                       ≡⟨⟩
@@ -600,7 +600,7 @@ module Opposite {ℓa ℓb : Level} where
       RawCategory.Object   (rawInv _) = Object
       RawCategory.Arrow    (rawInv _) = Arrow
       RawCategory.identity (rawInv _) = identity
-      RawCategory._∘_      (rawInv _) = _∘_
+      RawCategory._<<<_    (rawInv _) = _<<<_
 
     oppositeIsInvolution : opposite (opposite ℂ) ≡ ℂ
     oppositeIsInvolution = Category≡ rawInv

src/Cat/Category/Monad.agda

@@ -36,7 +36,7 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
   open Cat.Category.NaturalTransformation ℂ ℂ using (NaturalTransformation ; propIsNatural)
   private
     module ℂ = Category ℂ
-    open ℂ using (Object ; Arrow ; identity ; _∘_ ; _>>>_)
+    open ℂ using (Object ; Arrow ; identity ; _<<<_ ; _>>>_)
     module M = Monoidal ℂ
     module K = Kleisli  ℂ
 
@@ -74,21 +74,21 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
 
       backIsMonad : M.IsMonad backRaw
       M.IsMonad.isAssociative backIsMonad {X} = begin
-        joinT X  ∘ R.fmap (joinT X)  ≡⟨⟩
+        joinT X <<< R.fmap (joinT X)    ≡⟨⟩
-        join ∘ fmap (joinT X)         ≡⟨⟩
+        join <<< fmap (joinT X)         ≡⟨⟩
-        join ∘ fmap join              ≡⟨ isNaturalForeign ⟩
+        join <<< fmap join              ≡⟨ isNaturalForeign ⟩
-        join ∘ join                   ≡⟨⟩
+        join <<< join                   ≡⟨⟩
-        joinT X  ∘ joinT (R.omap X)  ∎
+        joinT X  <<< joinT (R.omap X)   ∎
       M.IsMonad.isInverse backIsMonad {X} = inv-l , inv-r
         where
         inv-l = begin
-          joinT X ∘ pureT (R.omap X) ≡⟨⟩
+          joinT X <<< pureT (R.omap X) ≡⟨⟩
-          join ∘ pure                 ≡⟨ fst isInverse ⟩
+          join <<< pure                ≡⟨ fst isInverse ⟩
-          identity                    ∎
+          identity                     ∎
         inv-r = begin
-          joinT X ∘ R.fmap (pureT X) ≡⟨⟩
+          joinT X <<< R.fmap (pureT X) ≡⟨⟩
-          join ∘ fmap pure            ≡⟨ snd isInverse ⟩
+          join <<< fmap pure           ≡⟨ snd isInverse ⟩
-          identity                    ∎
+          identity                     ∎
 
     back : K.Monad → M.Monad
     Monoidal.Monad.raw     (back m) = backRaw     m
@@ -101,11 +101,11 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
           → K.RawMonad.bind (forthRaw (backRaw m)) {X} {Y}
           ≡ K.RawMonad.bind (K.Monad.raw m)
         bindEq {X} {Y} = begin
-          K.RawMonad.bind (forthRaw (backRaw m)) ≡⟨⟩
+          K.RawMonad.bind (forthRaw (backRaw m))        ≡⟨⟩
-          (λ f → join ∘ fmap f)                  ≡⟨⟩
+          (λ f → join <<< fmap f)                       ≡⟨⟩
           (λ f → bind (f >>> pure) >>> bind identity)   ≡⟨ funExt lem ⟩
-          (λ f → bind f)                         ≡⟨⟩
+          (λ f → bind f)                                ≡⟨⟩
-          bind                                   ∎
+          bind                                          ∎
           where
           lem : (f : Arrow X (omap Y))
             → bind (f >>> pure) >>> bind identity
@@ -139,18 +139,18 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
 
         bindEq : ∀ {X Y} {f : Arrow X (Romap Y)} → KM.bind f ≡ bind f
         bindEq {X} {Y} {f} = begin
-          KM.bind f         ≡⟨⟩
+          KM.bind f           ≡⟨⟩
-          joinT Y ∘ Rfmap f ≡⟨⟩
+          joinT Y <<< Rfmap f ≡⟨⟩
-          bind f            ∎
+          bind f              ∎
 
         joinEq : ∀ {X} → KM.join ≡ joinT X
         joinEq {X} = begin
-          KM.join                ≡⟨⟩
+          KM.join                    ≡⟨⟩
-          KM.bind identity              ≡⟨⟩
+          KM.bind identity           ≡⟨⟩
-          bind identity                 ≡⟨⟩
+          bind identity              ≡⟨⟩
-          joinT X ∘ Rfmap identity      ≡⟨ cong (λ φ → _ ∘ φ) R.isIdentity ⟩
+          joinT X <<< Rfmap identity ≡⟨ cong (λ φ → _ <<< φ) R.isIdentity ⟩
-          joinT X ∘ identity            ≡⟨ ℂ.rightIdentity ⟩
+          joinT X <<< identity       ≡⟨ ℂ.rightIdentity ⟩
-          joinT X                ∎
+          joinT X                    ∎
 
         fmapEq : ∀ {A B} → KM.fmap {A} {B} ≡ Rfmap
         fmapEq {A} {B} = funExt (λ f → begin
@@ -160,8 +160,8 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
              Rfmap (f >>> pureT B) >>> joinT B     ≡⟨⟩
           Rfmap (f >>> pureT B) >>> joinT B        ≡⟨ cong (λ φ → φ >>> joinT B) R.isDistributive ⟩
           Rfmap f >>> Rfmap (pureT B) >>> joinT B  ≡⟨ ℂ.isAssociative ⟩
-          joinT B ∘ Rfmap (pureT B) ∘ Rfmap f      ≡⟨ cong (λ φ → φ ∘ Rfmap f) (snd isInverse) ⟩
+          joinT B <<< Rfmap (pureT B) <<< Rfmap f  ≡⟨ cong (λ φ → φ <<< Rfmap f) (snd isInverse) ⟩
-          identity ∘ Rfmap f                       ≡⟨ ℂ.leftIdentity ⟩
+          identity <<< Rfmap f                     ≡⟨ ℂ.leftIdentity ⟩
           Rfmap f                                  ∎
           )
 
@@ -183,8 +183,8 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
       joinTEq = funExt (λ X → begin
         M.RawMonad.joinT (backRaw (forth m)) X ≡⟨⟩
         KM.join ≡⟨⟩
-        joinT X ∘ Rfmap identity ≡⟨ cong (λ φ → joinT X ∘ φ) R.isIdentity ⟩
+        joinT X <<< Rfmap identity ≡⟨ cong (λ φ → joinT X <<< φ) R.isIdentity ⟩
-        joinT X ∘ identity ≡⟨ ℂ.rightIdentity ⟩
+        joinT X <<< identity ≡⟨ ℂ.rightIdentity ⟩
         joinT X ∎)
 
       joinNTEq : (λ i → NaturalTransformation F[ Req i ∘ Req i ] (Req i))

src/Cat/Category/Monad/Kleisli.agda

@@ -18,7 +18,7 @@ open import Cat.Category.NaturalTransformation ℂ ℂ
 private
   ℓ = ℓa ⊔ ℓb
   module ℂ = Category ℂ
-  open ℂ using (Arrow ; identity ; Object ; _∘_ ; _>>>_)
+  open ℂ using (Arrow ; identity ; Object ; _<<<_ ; _>>>_)
 
 -- | Data for a monad.
 --
@@ -34,7 +34,7 @@ record RawMonad : Set ℓ where
   --
   -- This should perhaps be defined in a "Klesli-version" of functors as well?
   fmap : ∀ {A B} → ℂ [ A , B ] → ℂ [ omap A , omap B ]
-  fmap f = bind (pure ∘ f)
+  fmap f = bind (pure <<< f)
 
   -- | Composition of monads aka. the kleisli-arrow.
   _>=>_ : {A B C : Object} → ℂ [ A , omap B ] → ℂ [ B , omap C ] → ℂ [ A , omap C ]
@@ -62,14 +62,14 @@ record RawMonad : Set ℓ where
   -- This is really a functor law. Should we have a kleisli-representation of
   -- functors as well and make them a super-class?
   Fusion = {X Y Z : Object} {g : ℂ [ Y , Z ]} {f : ℂ [ X , Y ]}
-    → fmap (g ∘ f) ≡ fmap g ∘ fmap f
+    → fmap (g <<< f) ≡ fmap g <<< fmap f
 
   -- In the ("foreign") formulation of a monad `IsNatural`'s analogue here would be:
   IsNaturalForeign : Set _
-  IsNaturalForeign = {X : Object} → join {X} ∘ fmap join ≡ join ∘ join
+  IsNaturalForeign = {X : Object} → join {X} <<< fmap join ≡ join <<< join
 
   IsInverse : Set _
-  IsInverse = {X : Object} → join {X} ∘ pure ≡ identity × join {X} ∘ fmap pure ≡ identity
+  IsInverse = {X : Object} → join {X} <<< pure ≡ identity × join {X} <<< fmap pure ≡ identity
 
 record IsMonad (raw : RawMonad) : Set ℓ where
   open RawMonad raw public
@@ -81,18 +81,21 @@ record IsMonad (raw : RawMonad) : Set ℓ where
   -- | Map fusion is admissable.
   fusion : Fusion
   fusion {g = g} {f} = begin
-    fmap (g ∘ f)               ≡⟨⟩
+    fmap (g <<< f)                 ≡⟨⟩
-    bind ((f >>> g) >>> pure)  ≡⟨ cong bind ℂ.isAssociative ⟩
+    bind ((f >>> g) >>> pure)      ≡⟨ cong bind ℂ.isAssociative ⟩
-    bind (f >>> (g >>> pure))  ≡⟨ cong (λ φ → bind (f >>> φ)) (sym (isNatural _)) ⟩
+    bind (f >>> (g >>> pure))
-    bind (f >>> (pure >>> (bind (g >>> pure)))) ≡⟨⟩
+      ≡⟨ cong (λ φ → bind (f >>> φ)) (sym (isNatural _)) ⟩
+    bind (f >>> (pure >>> (bind (g >>> pure))))
+      ≡⟨⟩
     bind (f >>> (pure >>> fmap g)) ≡⟨⟩
-    bind ((fmap g ∘ pure) ∘ f) ≡⟨ cong bind (sym ℂ.isAssociative) ⟩
+    bind ((fmap g <<< pure) <<< f) ≡⟨ cong bind (sym ℂ.isAssociative) ⟩
-    bind (fmap g ∘ (pure ∘ f)) ≡⟨ sym distrib ⟩
+    bind (fmap g <<< (pure <<< f)) ≡⟨ sym distrib ⟩
-    bind (pure ∘ g) ∘ bind (pure ∘ f) ≡⟨⟩
+    bind (pure <<< g) <<< bind (pure <<< f)
-    fmap g ∘ fmap f            ∎
+      ≡⟨⟩
+    fmap g <<< fmap f              ∎
     where
-      distrib : fmap g ∘ fmap f ≡ bind (fmap g ∘ (pure ∘ f))
+      distrib : fmap g <<< fmap f ≡ bind (fmap g <<< (pure <<< f))
-      distrib = isDistributive (pure ∘ g) (pure ∘ f)
+      distrib = isDistributive (pure <<< g) (pure <<< f)
 
   -- | This formulation gives rise to the following endo-functor.
   private
@@ -102,15 +105,15 @@ record IsMonad (raw : RawMonad) : Set ℓ where
 
     isFunctorR : IsFunctor ℂ ℂ rawR
     IsFunctor.isIdentity isFunctorR = begin
-      bind (pure ∘ identity) ≡⟨ cong bind (ℂ.rightIdentity) ⟩
+      bind (pure <<< identity) ≡⟨ cong bind (ℂ.rightIdentity) ⟩
-      bind pure              ≡⟨ isIdentity ⟩
+      bind pure                ≡⟨ isIdentity ⟩
-      identity               ∎
+      identity                 ∎
 
     IsFunctor.isDistributive isFunctorR {f = f} {g} = begin
-      bind (pure ∘ (g ∘ f))             ≡⟨⟩
+      bind (pure <<< (g <<< f))               ≡⟨⟩
-      fmap (g ∘ f)                      ≡⟨ fusion ⟩
+      fmap (g <<< f)                          ≡⟨ fusion ⟩
-      fmap g ∘ fmap f                   ≡⟨⟩
+      fmap g <<< fmap f                       ≡⟨⟩
-      bind (pure ∘ g) ∘ bind (pure ∘ f) ∎
+      bind (pure <<< g) <<< bind (pure <<< f) ∎
 
   -- FIXME Naming!
   R : EndoFunctor ℂ
@@ -129,20 +132,20 @@ record IsMonad (raw : RawMonad) : Set ℓ where
     pureT A = pure
     pureN : Natural R⁰ R pureT
     pureN {A} {B} f = begin
-      pureT B             ∘ R⁰.fmap f ≡⟨⟩
+      pureT B             <<< R⁰.fmap f ≡⟨⟩
-      pure            ∘ f          ≡⟨ sym (isNatural _) ⟩
+      pure            <<< f             ≡⟨ sym (isNatural _) ⟩
-      bind (pure ∘ f) ∘ pure       ≡⟨⟩
+      bind (pure <<< f) <<< pure        ≡⟨⟩
-      fmap f          ∘ pure       ≡⟨⟩
+      fmap f          <<< pure          ≡⟨⟩
-      R.fmap f       ∘ pureT A        ∎
+      R.fmap f       <<< pureT A        ∎
     joinT : Transformation R² R
     joinT C = join
     joinN : Natural R² R joinT
     joinN f = begin
-      join       ∘ R².fmap f  ≡⟨⟩
+      join       <<< R².fmap f                   ≡⟨⟩
-      bind identity     ∘ R².fmap f  ≡⟨⟩
+      bind identity     <<< R².fmap f            ≡⟨⟩
-      R².fmap f >>> bind identity    ≡⟨⟩
+      R².fmap f >>> bind identity                ≡⟨⟩
-      fmap (fmap f) >>> bind identity ≡⟨⟩
+      fmap (fmap f) >>> bind identity            ≡⟨⟩
-      fmap (bind (f >>> pure)) >>> bind identity          ≡⟨⟩
+      fmap (bind (f >>> pure)) >>> bind identity ≡⟨⟩
       bind (bind (f >>> pure) >>> pure) >>> bind identity
         ≡⟨ isDistributive _ _ ⟩
       bind ((bind (f >>> pure) >>> pure) >=> identity)
@@ -155,14 +158,14 @@ record IsMonad (raw : RawMonad) : Set ℓ where
         ≡⟨ cong bind ℂ.leftIdentity ⟩
       bind (bind (f >>> pure))
         ≡⟨ cong bind (sym ℂ.rightIdentity) ⟩
-      bind (identity >>> bind (f >>> pure)) ≡⟨⟩
+      bind (identity >>> bind (f >>> pure))   ≡⟨⟩
       bind (identity >=> (f >>> pure))
         ≡⟨ sym (isDistributive _ _) ⟩
-      bind identity     >>> bind (f >>> pure)    ≡⟨⟩
+      bind identity     >>> bind (f >>> pure) ≡⟨⟩
-      bind identity     >>> fmap f    ≡⟨⟩
+      bind identity     >>> fmap f            ≡⟨⟩
-      bind identity     >>> R.fmap f ≡⟨⟩
+      bind identity     >>> R.fmap f          ≡⟨⟩
-      R.fmap f  ∘ bind identity      ≡⟨⟩
+      R.fmap f  <<< bind identity             ≡⟨⟩
-      R.fmap f  ∘ join        ∎
+      R.fmap f  <<< join                      ∎
 
   pureNT : NaturalTransformation R⁰ R
   fst pureNT = pureT

src/Cat/Category/Monad/Monoidal.agda

@@ -16,7 +16,7 @@ module Cat.Category.Monad.Monoidal {ℓa ℓb : Level} (ℂ : Category ℓa ℓb
 private
   ℓ = ℓa ⊔ ℓb
 
-open Category ℂ using (Object ; Arrow ; identity ; _∘_)
+open Category ℂ using (Object ; Arrow ; identity ; _<<<_)
 open import Cat.Category.NaturalTransformation ℂ ℂ
   using (NaturalTransformation ; Transformation ; Natural)
 
@@ -42,19 +42,19 @@ record RawMonad : Set ℓ where
   Rfmap = Functor.fmap R
 
   bind : {X Y : Object} → ℂ [ X , Romap Y ] → ℂ [ Romap X , Romap Y ]
-  bind {X} {Y} f = joinT Y ∘ Rfmap f
+  bind {X} {Y} f = joinT Y <<< Rfmap f
 
   IsAssociative : Set _
   IsAssociative = {X : Object}
-    → joinT X ∘ Rfmap (joinT X) ≡ joinT X ∘ joinT (Romap X)
+    → joinT X <<< Rfmap (joinT X) ≡ joinT X <<< joinT (Romap X)
   IsInverse : Set _
   IsInverse = {X : Object}
-    → joinT X ∘ pureT (Romap X) ≡ identity
+    → joinT X <<< pureT (Romap X) ≡ identity
-    × joinT X ∘ Rfmap (pureT X) ≡ identity
+    × joinT X <<< Rfmap (pureT X) ≡ identity
-  IsNatural = ∀ {X Y} f → joinT Y ∘ Rfmap f ∘ pureT X ≡ f
+  IsNatural = ∀ {X Y} f → joinT Y <<< Rfmap f <<< pureT X ≡ f
   IsDistributive = ∀ {X Y Z} (g : Arrow Y (Romap Z)) (f : Arrow X (Romap Y))
-    → joinT Z ∘ Rfmap g ∘ (joinT Y ∘ Rfmap f)
+    → joinT Z <<< Rfmap g <<< (joinT Y <<< Rfmap f)
-    ≡ joinT Z ∘ Rfmap (joinT Z ∘ Rfmap g ∘ f)
+    ≡ joinT Z <<< Rfmap (joinT Z <<< Rfmap g <<< f)
 
 record IsMonad (raw : RawMonad) : Set ℓ where
   open RawMonad raw public
@@ -68,48 +68,48 @@ record IsMonad (raw : RawMonad) : Set ℓ where
 
   isNatural : IsNatural
   isNatural {X} {Y} f = begin
-    joinT Y ∘ R.fmap f ∘ pureT X     ≡⟨ sym ℂ.isAssociative ⟩
+    joinT Y <<< R.fmap f <<< pureT X     ≡⟨ sym ℂ.isAssociative ⟩
-    joinT Y ∘ (R.fmap f ∘ pureT X)   ≡⟨ cong (λ φ → joinT Y ∘ φ) (sym (pureN f)) ⟩
+    joinT Y <<< (R.fmap f <<< pureT X)   ≡⟨ cong (λ φ → joinT Y <<< φ) (sym (pureN f)) ⟩
-    joinT Y ∘ (pureT (R.omap Y) ∘ f) ≡⟨ ℂ.isAssociative ⟩
+    joinT Y <<< (pureT (R.omap Y) <<< f) ≡⟨ ℂ.isAssociative ⟩
-    joinT Y ∘ pureT (R.omap Y) ∘ f   ≡⟨ cong (λ φ → φ ∘ f) (fst isInverse) ⟩
+    joinT Y <<< pureT (R.omap Y) <<< f   ≡⟨ cong (λ φ → φ <<< f) (fst isInverse) ⟩
-    identity ∘ f                     ≡⟨ ℂ.leftIdentity ⟩
+    identity <<< f                       ≡⟨ ℂ.leftIdentity ⟩
-    f                                ∎
+    f                                    ∎
 
   isDistributive : IsDistributive
   isDistributive {X} {Y} {Z} g f = sym aux
     where
     module R² = Functor F[ R ∘ R ]
     distrib3 : ∀ {A B C D} {a : Arrow C D} {b : Arrow B C} {c : Arrow A B}
-      → R.fmap (a ∘ b ∘ c)
+      → R.fmap (a <<< b <<< c)
-      ≡ R.fmap a ∘ R.fmap b ∘ R.fmap c
+      ≡ R.fmap a <<< R.fmap b <<< R.fmap c
     distrib3 {a = a} {b} {c} = begin
-      R.fmap (a ∘ b ∘ c)               ≡⟨ R.isDistributive ⟩
+      R.fmap (a <<< b <<< c)              ≡⟨ R.isDistributive ⟩
-      R.fmap (a ∘ b) ∘ R.fmap c       ≡⟨ cong (_∘ _) R.isDistributive ⟩
+      R.fmap (a <<< b) <<< R.fmap c       ≡⟨ cong (_<<< _) R.isDistributive ⟩
-      R.fmap a ∘ R.fmap b ∘ R.fmap c ∎
+      R.fmap a <<< R.fmap b <<< R.fmap c  ∎
     aux = begin
-      joinT Z ∘ R.fmap (joinT Z ∘ R.fmap g ∘ f)
+      joinT Z <<< R.fmap (joinT Z <<< R.fmap g <<< f)
-        ≡⟨ cong (λ φ → joinT Z ∘ φ) distrib3 ⟩
+        ≡⟨ cong (λ φ → joinT Z <<< φ) distrib3 ⟩
-      joinT Z ∘ (R.fmap (joinT Z) ∘ R.fmap (R.fmap g) ∘ R.fmap f)
+      joinT Z <<< (R.fmap (joinT Z) <<< R.fmap (R.fmap g) <<< R.fmap f)
         ≡⟨⟩
-      joinT Z ∘ (R.fmap (joinT Z) ∘ R².fmap g ∘ R.fmap f)
+      joinT Z <<< (R.fmap (joinT Z) <<< R².fmap g <<< R.fmap f)
-        ≡⟨ cong (_∘_ (joinT Z)) (sym ℂ.isAssociative) ⟩
+        ≡⟨ cong (_<<<_ (joinT Z)) (sym ℂ.isAssociative) ⟩
-      joinT Z ∘ (R.fmap (joinT Z) ∘ (R².fmap g ∘ R.fmap f))
+      joinT Z <<< (R.fmap (joinT Z) <<< (R².fmap g <<< R.fmap f))
         ≡⟨ ℂ.isAssociative ⟩
-      (joinT Z ∘ R.fmap (joinT Z)) ∘ (R².fmap g ∘ R.fmap f)
+      (joinT Z <<< R.fmap (joinT Z)) <<< (R².fmap g <<< R.fmap f)
-        ≡⟨ cong (λ φ → φ ∘ (R².fmap g ∘ R.fmap f)) isAssociative ⟩
+        ≡⟨ cong (λ φ → φ <<< (R².fmap g <<< R.fmap f)) isAssociative ⟩
-      (joinT Z ∘ joinT (R.omap Z)) ∘ (R².fmap g ∘ R.fmap f)
+      (joinT Z <<< joinT (R.omap Z)) <<< (R².fmap g <<< R.fmap f)
         ≡⟨ ℂ.isAssociative ⟩
-      joinT Z ∘ joinT (R.omap Z) ∘ R².fmap g ∘ R.fmap f
+      joinT Z <<< joinT (R.omap Z) <<< R².fmap g <<< R.fmap f
         ≡⟨⟩
-      ((joinT Z ∘ joinT (R.omap Z)) ∘ R².fmap g) ∘ R.fmap f
+      ((joinT Z <<< joinT (R.omap Z)) <<< R².fmap g) <<< R.fmap f
-        ≡⟨ cong (_∘ R.fmap f) (sym ℂ.isAssociative) ⟩
+        ≡⟨ cong (_<<< R.fmap f) (sym ℂ.isAssociative) ⟩
-      (joinT Z ∘ (joinT (R.omap Z) ∘ R².fmap g)) ∘ R.fmap f
+      (joinT Z <<< (joinT (R.omap Z) <<< R².fmap g)) <<< R.fmap f
-        ≡⟨ cong (λ φ → φ ∘ R.fmap f) (cong (_∘_ (joinT Z)) (joinN g)) ⟩
+        ≡⟨ cong (λ φ → φ <<< R.fmap f) (cong (_<<<_ (joinT Z)) (joinN g)) ⟩
-      (joinT Z ∘ (R.fmap g ∘ joinT Y)) ∘ R.fmap f
+      (joinT Z <<< (R.fmap g <<< joinT Y)) <<< R.fmap f
-        ≡⟨ cong (_∘ R.fmap f) ℂ.isAssociative ⟩
+        ≡⟨ cong (_<<< R.fmap f) ℂ.isAssociative ⟩
-      joinT Z ∘ R.fmap g ∘ joinT Y ∘ R.fmap f
+      joinT Z <<< R.fmap g <<< joinT Y <<< R.fmap f
         ≡⟨ sym (Category.isAssociative ℂ) ⟩
-      joinT Z ∘ R.fmap g ∘ (joinT Y ∘ R.fmap f)
+      joinT Z <<< R.fmap g <<< (joinT Y <<< R.fmap f)
         ∎
 
 record Monad : Set ℓ where

src/Cat/Category/Product.agda

@@ -105,8 +105,8 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
           × ℂ [ y1 ∘ f ] ≡ x1
           }
       ; identity = λ{ {X , f , g} → ℂ.identity {X} , ℂ.rightIdentity , ℂ.rightIdentity}
-      ; _∘_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1} (f , f0 , f1) (g , g0 , g1)
+      ; _<<<_ = λ { {_ , a0 , a1} {_ , b0 , b1} {_ , c0 , c1} (f , f0 , f1) (g , g0 , g1)
-        → (f ℂ.∘ g)
+        → (f ℂ.<<< g)
           , (begin
               ℂ [ c0 ∘ ℂ [ f ∘ g ] ] ≡⟨ ℂ.isAssociative ⟩
               ℂ [ ℂ [ c0 ∘ f ] ∘ g ] ≡⟨ cong (λ φ → ℂ [ φ ∘ g ]) f0 ⟩
@@ -134,9 +134,9 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
         isAssociative {A'@(A , a0 , a1)} {B , _} {C , c0 , c1} {D'@(D , d0 , d1)} {ff@(f , f0 , f1)} {gg@(g , g0 , g1)} {hh@(h , h0 , h1)} i
           = s0 i , lemPropF propEqs s0 {P.snd l} {P.snd r} i
           where
-          l = hh ∘ (gg ∘ ff)
+          l = hh <<< (gg <<< ff)
-          r = hh ∘ gg ∘ ff
+          r = hh <<< gg <<< ff
-          -- s0 : h ℂ.∘ (g ℂ.∘ f) ≡ h ℂ.∘ g ℂ.∘ f
+          -- s0 : h ℂ.<<< (g ℂ.<<< f) ≡ h ℂ.<<< g ℂ.<<< f
           s0 : fst l ≡ fst r
           s0 = ℂ.isAssociative {f = f} {g} {h}
 
@@ -144,18 +144,18 @@ module Try0 {ℓa ℓb : Level} {ℂ : Category ℓa ℓb}
         isIdentity : IsIdentity identity
         isIdentity {AA@(A , a0 , a1)} {BB@(B , b0 , b1)} {f , f0 , f1} = leftIdentity , rightIdentity
           where
-          leftIdentity : identity ∘ (f , f0 , f1) ≡ (f , f0 , f1)
+          leftIdentity : identity <<< (f , f0 , f1) ≡ (f , f0 , f1)
           leftIdentity i = l i , lemPropF propEqs l {snd L} {snd R} i
             where
-            L = identity ∘ (f , f0 , f1)
+            L = identity <<< (f , f0 , f1)
             R : Arrow AA BB
             R = f , f0 , f1
             l : fst L ≡ fst R
             l = ℂ.leftIdentity
-          rightIdentity : (f , f0 , f1) ∘ identity ≡ (f , f0 , f1)
+          rightIdentity : (f , f0 , f1) <<< identity ≡ (f , f0 , f1)
           rightIdentity i = l i , lemPropF propEqs l {snd L} {snd R} i
             where
-            L = (f , f0 , f1) ∘ identity
+            L = (f , f0 , f1) <<< identity
             R : Arrow AA BB
             R = (f , f0 , f1)
             l : ℂ [ f ∘ ℂ.identity ] ≡ f