# Opposite categories

Content created by Fredrik Bakke.

Created on 2023-11-01.

module category-theory.opposite-categories where

Imports
open import category-theory.categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.opposite-precategories
open import category-theory.precategories

open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.identity-types
open import foundation.involutions
open import foundation.sets
open import foundation.strictly-involutive-identity-types
open import foundation.subtypes
open import foundation.universe-levels


## Idea

Let C be a category, its opposite category Cᵒᵖ is given by reversing every morphism.

## Lemma

### The underlying precategory is a category if and only if the opposite is a category

abstract
is-category-opposite-is-category-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2) →
is-category-Precategory C →
is-category-Precategory (opposite-Precategory C)
is-category-opposite-is-category-Precategory C is-category-C x y =
is-equiv-htpy-equiv
( compute-iso-opposite-Precategory C ∘e
equiv-inv-iso-Precategory C ∘e
(_ , is-category-C x y))
( λ where
refl →
eq-type-subtype
( is-iso-prop-Precategory (opposite-Precategory C))
( refl))

abstract
is-category-is-category-opposite-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2) →
is-category-Precategory (opposite-Precategory C) →
is-category-Precategory C
is-category-is-category-opposite-Precategory C =
is-category-opposite-is-category-Precategory (opposite-Precategory C)


## Definitions

### The opposite category

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

obj-opposite-Category : UU l1
obj-opposite-Category = obj-opposite-Precategory (precategory-Category C)

hom-set-opposite-Category : (x y : obj-opposite-Category) → Set l2
hom-set-opposite-Category =
hom-set-opposite-Precategory (precategory-Category C)

hom-opposite-Category : (x y : obj-opposite-Category) → UU l2
hom-opposite-Category = hom-opposite-Precategory (precategory-Category C)

comp-hom-opposite-Category :
{x y z : obj-opposite-Category} →
hom-opposite-Category y z →
hom-opposite-Category x y →
hom-opposite-Category x z
comp-hom-opposite-Category =
comp-hom-opposite-Precategory (precategory-Category C)

associative-comp-hom-opposite-Category :
{x y z w : obj-opposite-Category}
(h : hom-opposite-Category z w)
(g : hom-opposite-Category y z)
(f : hom-opposite-Category x y) →
comp-hom-opposite-Category (comp-hom-opposite-Category h g) f ＝
comp-hom-opposite-Category h (comp-hom-opposite-Category g f)
associative-comp-hom-opposite-Category =
associative-comp-hom-opposite-Precategory (precategory-Category C)

involutive-eq-associative-comp-hom-opposite-Category :
{x y z w : obj-opposite-Category}
(h : hom-opposite-Category z w)
(g : hom-opposite-Category y z)
(f : hom-opposite-Category x y) →
comp-hom-opposite-Category (comp-hom-opposite-Category h g) f ＝ⁱ
comp-hom-opposite-Category h (comp-hom-opposite-Category g f)
involutive-eq-associative-comp-hom-opposite-Category =
involutive-eq-associative-comp-hom-opposite-Precategory
( precategory-Category C)

id-hom-opposite-Category :
{x : obj-opposite-Category} → hom-opposite-Category x x
id-hom-opposite-Category =
id-hom-opposite-Precategory (precategory-Category C)

left-unit-law-comp-hom-opposite-Category :
{x y : obj-opposite-Category}
(f : hom-opposite-Category x y) →
comp-hom-opposite-Category id-hom-opposite-Category f ＝ f
left-unit-law-comp-hom-opposite-Category =
left-unit-law-comp-hom-opposite-Precategory (precategory-Category C)

right-unit-law-comp-hom-opposite-Category :
{x y : obj-opposite-Category} (f : hom-opposite-Category x y) →
comp-hom-opposite-Category f id-hom-opposite-Category ＝ f
right-unit-law-comp-hom-opposite-Category =
right-unit-law-comp-hom-opposite-Precategory (precategory-Category C)

precategory-opposite-Category : Precategory l1 l2
precategory-opposite-Category = opposite-Precategory (precategory-Category C)

opposite-Category : Category l1 l2
pr1 opposite-Category = precategory-opposite-Category
pr2 opposite-Category =
is-category-opposite-is-category-Precategory
( precategory-Category C)
( is-category-Category C)


## Properties

### The opposite construction is an involution on the type of categories

is-involution-opposite-Category :
{l1 l2 : Level} → is-involution (opposite-Category {l1} {l2})
is-involution-opposite-Category C =
eq-type-subtype
( is-category-prop-Precategory)
( is-involution-opposite-Precategory (precategory-Category C))

involution-opposite-Category :
(l1 l2 : Level) → involution (Category l1 l2)
pr1 (involution-opposite-Category l1 l2) = opposite-Category
pr2 (involution-opposite-Category l1 l2) = is-involution-opposite-Category

is-equiv-opposite-Category :
{l1 l2 : Level} → is-equiv (opposite-Category {l1} {l2})
is-equiv-opposite-Category =
is-equiv-is-involution is-involution-opposite-Category

equiv-opposite-Category :
(l1 l2 : Level) → Category l1 l2 ≃ Category l1 l2
equiv-opposite-Category l1 l2 =
equiv-involution (involution-opposite-Category l1 l2)