Natural transformations between functors between categories

Content created by Fredrik Bakke.

Created on 2023-09-27.
Last modified on 2023-11-01.

module category-theory.natural-transformations-functors-categories where
Imports
open import category-theory.categories
open import category-theory.functors-categories
open import category-theory.natural-transformations-functors-precategories

open import foundation.embeddings
open import foundation.equivalences
open import foundation.homotopies
open import foundation.identity-types
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

Idea

A natural transformation between functors between categories is a natural transformation between the functors on the underlying precategories.

Definition

module _
  {l1 l2 l3 l4 : Level}
  (C : Category l1 l2)
  (D : Category l3 l4)
  (F G : functor-Category C D)
  where

  hom-family-functor-Category : UU (l1  l4)
  hom-family-functor-Category =
    hom-family-functor-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  is-natural-transformation-Category :
    hom-family-functor-Category  UU (l1  l2  l4)
  is-natural-transformation-Category =
    is-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  natural-transformation-Category : UU (l1  l2  l4)
  natural-transformation-Category =
    natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  hom-family-natural-transformation-Category :
    natural-transformation-Category  hom-family-functor-Category
  hom-family-natural-transformation-Category =
    hom-family-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  naturality-natural-transformation-Category :
    (γ : natural-transformation-Category) 
    is-natural-transformation-Category
      ( hom-family-natural-transformation-Category γ)
  naturality-natural-transformation-Category =
    naturality-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

Composition and identity of natural transformations

module _
  {l1 l2 l3 l4 : Level} (C : Category l1 l2) (D : Category l3 l4)
  where

  id-natural-transformation-Category :
    (F : functor-Category C D)  natural-transformation-Category C D F F
  id-natural-transformation-Category =
    id-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)

  comp-natural-transformation-Category :
    (F G H : functor-Category C D) 
    natural-transformation-Category C D G H 
    natural-transformation-Category C D F G 
    natural-transformation-Category C D F H
  comp-natural-transformation-Category =
    comp-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)

Properties

That a family of morphisms is a natural transformation is a proposition

This follows from the fact that the hom-types are sets.

module _
  {l1 l2 l3 l4 : Level}
  (C : Category l1 l2)
  (D : Category l3 l4)
  (F G : functor-Category C D)
  where

  is-prop-is-natural-transformation-Category :
    ( γ : hom-family-functor-Category C D F G) 
    is-prop (is-natural-transformation-Category C D F G γ)
  is-prop-is-natural-transformation-Category =
    is-prop-is-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  is-natural-transformation-prop-Category :
    ( γ : hom-family-functor-Category C D F G)  Prop (l1  l2  l4)
  is-natural-transformation-prop-Category =
    is-natural-transformation-prop-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

The set of natural transformations

  is-emb-hom-family-natural-transformation-Category :
    is-emb (hom-family-natural-transformation-Category C D F G)
  is-emb-hom-family-natural-transformation-Category =
    is-emb-hom-family-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  is-set-natural-transformation-Category :
    is-set (natural-transformation-Category C D F G)
  is-set-natural-transformation-Category =
    is-set-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  natural-transformation-set-Category : Set (l1  l2  l4)
  natural-transformation-set-Category =
    natural-transformation-set-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  extensionality-natural-transformation-Category :
    (α β : natural-transformation-Category C D F G) 
    ( α  β) 
    ( hom-family-natural-transformation-Category C D F G α ~
      hom-family-natural-transformation-Category C D F G β)
  extensionality-natural-transformation-Category =
    extensionality-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

  eq-htpy-hom-family-natural-transformation-Category :
    (α β : natural-transformation-Category C D F G) 
    ( hom-family-natural-transformation-Category C D F G α ~
      hom-family-natural-transformation-Category C D F G β) 
    α  β
  eq-htpy-hom-family-natural-transformation-Category =
    eq-htpy-hom-family-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      ( F)
      ( G)

Categorical laws for natural transformations

module _
  {l1 l2 l3 l4 : Level} (C : Category l1 l2) (D : Category l3 l4)
  where

  right-unit-law-comp-natural-transformation-Category :
    {F G : functor-Category C D}
    (α : natural-transformation-Category C D F G) 
    comp-natural-transformation-Category C D F F G α
      ( id-natural-transformation-Category C D F)  α
  right-unit-law-comp-natural-transformation-Category {F} {G} =
    right-unit-law-comp-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      F G

  left-unit-law-comp-natural-transformation-Category :
    {F G : functor-Category C D}
    (α : natural-transformation-Category C D F G) 
    comp-natural-transformation-Category C D F G G
      ( id-natural-transformation-Category C D G) α  α
  left-unit-law-comp-natural-transformation-Category {F} {G} =
    left-unit-law-comp-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)
      F G

  associative-comp-natural-transformation-Category :
    (F G H I : functor-Category C D)
    (α : natural-transformation-Category C D F G)
    (β : natural-transformation-Category C D G H)
    (γ : natural-transformation-Category C D H I) 
    comp-natural-transformation-Category C D F G I
      ( comp-natural-transformation-Category C D G H I γ β) α 
    comp-natural-transformation-Category C D F H I γ
      ( comp-natural-transformation-Category C D F G H β α)
  associative-comp-natural-transformation-Category =
    associative-comp-natural-transformation-Precategory
      ( precategory-Category C)
      ( precategory-Category D)

Recent changes