Refactor the definition of monads (#1019)

This PR breaks up the definition of a monad into smaller parts.
UniMath · Feb 8, 2024 · 8a8dd0a · 8a8dd0a
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diff --git a/src/category-theory.lagda.md b/src/category-theory.lagda.md
@@ -120,6 +120,8 @@ open import category-theory.opposite-categories public
 open import category-theory.opposite-large-precategories public
 open import category-theory.opposite-precategories public
 open import category-theory.opposite-preunivalent-categories public
+open import category-theory.pointed-endofunctors-categories public
+open import category-theory.pointed-endofunctors-precategories public
 open import category-theory.precategories public
 open import category-theory.precategory-of-elements-of-a-presheaf public
 open import category-theory.precategory-of-functors public

diff --git a/src/category-theory/monads-on-categories.lagda.md b/src/category-theory/monads-on-categories.lagda.md
@@ -8,23 +8,240 @@ module category-theory.monads-on-categories where
 
 ```agda
 open import category-theory.categories
+open import category-theory.functors-categories
 open import category-theory.monads-on-precategories
-open import category-theory.precategories
+open import category-theory.natural-transformations-functors-categories
+open import category-theory.pointed-endofunctors-categories
 
+open import foundation.dependent-pair-types
+open import foundation.identity-types
 open import foundation.universe-levels
+
+open import foundation-core.cartesian-product-types
 ```
 
 </details>
 
+## Idea
+
+A {{#concept "monad" Disambiguation="on a category" Agda=monad-Category}} on a
+[category](category-theory.categories.md) `C` consists of an
+[endofunctor](category-theory.functors-categories.md) `T : C → C` together with
+two
+[natural transformations](category-theory.natural-transformations-functors-categories.md):
+`η : 1_C ⇒ T` and `μ : T² ⇒ T`, where `1_C : C → C` is the identity functor for
+`C`, and `T²` is the functor `T ∘ T : C → C`. These must satisfy the following
+{{#concept "monad laws" Disambiguation="monad on a category"}}:
+
+- **Associativity:** `μ ∘ (T • μ) = μ ∘ (μ • T)`
+- The **left unit law:** `μ ∘ (T • η) = 1_T`
+- The **right unit law:** `μ ∘ (η • T) = 1_T`.
+
+Here, `•` denotes
+[whiskering](category-theory.natural-transformations-functors-precategories.md#whiskering),
+and `1_T : T ⇒ T` denotes the identity natural transformation for `T`.
+
 ## Definitions
 
+### Multiplication structure on a pointed endofunctor on a category
+
+```agda
+module _
+  {l1 l2 : Level} (C : Category l1 l2)
+  (T : pointed-endofunctor-Category C)
+  where
+
+  structure-multiplication-pointed-endofunctor-Category : UU (l1 ⊔ l2)
+  structure-multiplication-pointed-endofunctor-Category =
+    structure-multiplication-pointed-endofunctor-Precategory
+      ( precategory-Category C)
+      ( T)
+```
+
+### Associativity of multiplication on a pointed endofunctor on a category
+
+```agda
+module _
+  {l1 l2 : Level} (C : Category l1 l2)
+  (T : pointed-endofunctor-Category C)
+  (μ : structure-multiplication-pointed-endofunctor-Category C T)
+  where
+
+  associative-mul-pointed-endofunctor-Category : UU (l1 ⊔ l2)
+  associative-mul-pointed-endofunctor-Category =
+    associative-mul-pointed-endofunctor-Precategory
+      ( precategory-Category C)
+      ( T)
+      ( μ)
+```
+
+### The left unit law on a multiplication on a pointed endofunctor
+
+```agda
+module _
+  {l1 l2 : Level} (C : Category l1 l2)
+  (T : pointed-endofunctor-Category C)
+  (μ : structure-multiplication-pointed-endofunctor-Category C T)
+  where
+
+  left-unit-law-mul-pointed-endofunctor-Category : UU (l1 ⊔ l2)
+  left-unit-law-mul-pointed-endofunctor-Category =
+    left-unit-law-mul-pointed-endofunctor-Precategory
+      ( precategory-Category C)
+      ( T)
+      ( μ)
+```
+
+### The right unit law on a multiplication on a pointed endofunctor
+
+```agda
+module _
+  {l1 l2 : Level} (C : Category l1 l2)
+  (T : pointed-endofunctor-Category C)
+  (μ : structure-multiplication-pointed-endofunctor-Category C T)
+  where
+
+  right-unit-law-mul-pointed-endofunctor-Category : UU (l1 ⊔ l2)
+  right-unit-law-mul-pointed-endofunctor-Category =
+    right-unit-law-mul-pointed-endofunctor-Precategory
+      ( precategory-Category C)
+      ( T)
+      ( μ)
+```
+
+### The structure of a monad on a pointed endofunctor on a category
+
+```agda
+module _
+  {l1 l2 : Level} (C : Category l1 l2)
+  (T : pointed-endofunctor-Category C)
+  where
+
+  structure-monad-pointed-endofunctor-Category : UU (l1 ⊔ l2)
+  structure-monad-pointed-endofunctor-Category =
+    structure-monad-pointed-endofunctor-Precategory
+      ( precategory-Category C)
+      ( T)
+```
+
 ### The type of monads on categories
 
 ```agda
 module _
-  {l : Level} (C : Category l l)
+  {l1 l2 : Level} (C : Category l1 l2)
   where
 
-  monad-Category : UU l
-  monad-Category = monad-Precategory l (precategory-Category C)
+  monad-Category : UU (l1 ⊔ l2)
+  monad-Category = monad-Precategory (precategory-Category C)
+
+  module _
+    (T : monad-Category)
+    where
+
+    pointed-endofunctor-monad-Category :
+      pointed-endofunctor-Category C
+    pointed-endofunctor-monad-Category =
+      pointed-endofunctor-monad-Precategory (precategory-Category C) T
+
+    endofunctor-monad-Category :
+      functor-Category C C
+    endofunctor-monad-Category =
+      endofunctor-monad-Precategory (precategory-Category C) T
+
+    obj-endofunctor-monad-Category :
+      obj-Category C → obj-Category C
+    obj-endofunctor-monad-Category =
+      obj-endofunctor-monad-Precategory (precategory-Category C) T
+
+    hom-endofunctor-monad-Category :
+      {X Y : obj-Category C} →
+      hom-Category C X Y →
+      hom-Category C
+        ( obj-endofunctor-monad-Category X)
+        ( obj-endofunctor-monad-Category Y)
+    hom-endofunctor-monad-Category =
+      hom-endofunctor-monad-Precategory (precategory-Category C) T
+
+    preserves-id-endofunctor-monad-Category :
+      (X : obj-Category C) →
+      hom-endofunctor-monad-Category (id-hom-Category C {X}) ＝
+      id-hom-Category C
+    preserves-id-endofunctor-monad-Category =
+      preserves-id-endofunctor-monad-Precategory (precategory-Category C) T
+
+    preserves-comp-endofunctor-monad-Category :
+      {X Y Z : obj-Category C} →
+      (g : hom-Category C Y Z) (f : hom-Category C X Y) →
+      hom-endofunctor-monad-Category (comp-hom-Category C g f) ＝
+      comp-hom-Category C
+        ( hom-endofunctor-monad-Category g)
+        ( hom-endofunctor-monad-Category f)
+    preserves-comp-endofunctor-monad-Category =
+      preserves-comp-endofunctor-monad-Precategory (precategory-Category C) T
+
+    unit-monad-Category :
+      pointing-endofunctor-Category C endofunctor-monad-Category
+    unit-monad-Category =
+      unit-monad-Precategory (precategory-Category C) T
+
+    hom-unit-monad-Category :
+      hom-family-functor-Category C C
+        ( id-functor-Category C)
+        ( endofunctor-monad-Category)
+    hom-unit-monad-Category =
+      hom-unit-monad-Precategory (precategory-Category C) T
+
+    naturality-unit-monad-Category :
+      is-natural-transformation-Category C C
+        ( id-functor-Category C)
+        ( endofunctor-monad-Category)
+        ( hom-unit-monad-Category)
+    naturality-unit-monad-Category =
+      naturality-unit-monad-Precategory (precategory-Category C) T
+
+    mul-monad-Category :
+      structure-multiplication-pointed-endofunctor-Category C
+        ( pointed-endofunctor-monad-Category)
+    mul-monad-Category =
+      mul-monad-Precategory (precategory-Category C) T
+
+    hom-mul-monad-Category :
+      hom-family-functor-Category C C
+        ( comp-functor-Category C C C
+          ( endofunctor-monad-Category)
+          ( endofunctor-monad-Category))
+        ( endofunctor-monad-Category)
+    hom-mul-monad-Category =
+      hom-mul-monad-Precategory (precategory-Category C) T
+
+    naturality-mul-monad-Category :
+      is-natural-transformation-Category C C
+        ( comp-functor-Category C C C
+          ( endofunctor-monad-Category)
+          ( endofunctor-monad-Category))
+        ( endofunctor-monad-Category)
+        ( hom-mul-monad-Category)
+    naturality-mul-monad-Category =
+      naturality-mul-monad-Precategory (precategory-Category C) T
+
+    associative-mul-monad-Category :
+      associative-mul-pointed-endofunctor-Category C
+        ( pointed-endofunctor-monad-Category)
+        ( mul-monad-Category)
+    associative-mul-monad-Category =
+      associative-mul-monad-Precategory (precategory-Category C) T
+
+    left-unit-law-mul-monad-Category :
+      left-unit-law-mul-pointed-endofunctor-Category C
+        ( pointed-endofunctor-monad-Category)
+        ( mul-monad-Category)
+    left-unit-law-mul-monad-Category =
+      left-unit-law-mul-monad-Precategory (precategory-Category C) T
+
+    right-unit-law-mul-monad-Category :
+      right-unit-law-mul-pointed-endofunctor-Category C
+        ( pointed-endofunctor-monad-Category)
+        ( mul-monad-Category)
+    right-unit-law-mul-monad-Category =
+      right-unit-law-mul-monad-Precategory (precategory-Category C) T
 ```