# Isomorphisms in large precategories

Content created by Fredrik Bakke, Egbert Rijke and Gregor Perčič.

Created on 2023-09-13.

module category-theory.isomorphisms-in-large-precategories where

Imports
open import category-theory.isomorphisms-in-precategories
open import category-theory.large-precategories

open import foundation.action-on-identifications-functions
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.propositions
open import foundation.retractions
open import foundation.sections
open import foundation.sets
open import foundation.subtypes
open import foundation.universe-levels


## Idea

An isomorphism in a large precategory C is a morphism f : X → Y in C for which there exists a morphism g : Y → X such that f ∘ g ＝ id and g ∘ f ＝ id.

## Definitions

### The predicate of being an isomorphism

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : hom-Large-Precategory C X Y)
where

is-iso-Large-Precategory : UU (β l1 l1 ⊔ β l2 l1 ⊔ β l2 l2)
is-iso-Large-Precategory =
Σ ( hom-Large-Precategory C Y X)
( λ g →
( comp-hom-Large-Precategory C f g ＝ id-hom-Large-Precategory C) ×
( comp-hom-Large-Precategory C g f ＝ id-hom-Large-Precategory C))

hom-inv-is-iso-Large-Precategory :
is-iso-Large-Precategory → hom-Large-Precategory C Y X
hom-inv-is-iso-Large-Precategory = pr1

is-section-hom-inv-is-iso-Large-Precategory :
(H : is-iso-Large-Precategory) →
comp-hom-Large-Precategory C f (hom-inv-is-iso-Large-Precategory H) ＝
id-hom-Large-Precategory C
is-section-hom-inv-is-iso-Large-Precategory = pr1 ∘ pr2

is-retraction-hom-inv-is-iso-Large-Precategory :
(H : is-iso-Large-Precategory) →
comp-hom-Large-Precategory C (hom-inv-is-iso-Large-Precategory H) f ＝
id-hom-Large-Precategory C
is-retraction-hom-inv-is-iso-Large-Precategory = pr2 ∘ pr2


### Isomorphisms in a large precategory

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
(X : obj-Large-Precategory C l1) (Y : obj-Large-Precategory C l2)
where

iso-Large-Precategory : UU (β l1 l1 ⊔ β l1 l2 ⊔ β l2 l1 ⊔ β l2 l2)
iso-Large-Precategory =
Σ (hom-Large-Precategory C X Y) (is-iso-Large-Precategory C)

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : iso-Large-Precategory C X Y)
where

hom-iso-Large-Precategory : hom-Large-Precategory C X Y
hom-iso-Large-Precategory = pr1 f

is-iso-iso-Large-Precategory :
is-iso-Large-Precategory C hom-iso-Large-Precategory
is-iso-iso-Large-Precategory = pr2 f

hom-inv-iso-Large-Precategory : hom-Large-Precategory C Y X
hom-inv-iso-Large-Precategory = pr1 (pr2 f)

is-section-hom-inv-iso-Large-Precategory :
( comp-hom-Large-Precategory C
( hom-iso-Large-Precategory)
( hom-inv-iso-Large-Precategory)) ＝
( id-hom-Large-Precategory C)
is-section-hom-inv-iso-Large-Precategory = pr1 (pr2 (pr2 f))

is-retraction-hom-inv-iso-Large-Precategory :
( comp-hom-Large-Precategory C
( hom-inv-iso-Large-Precategory)
( hom-iso-Large-Precategory)) ＝
( id-hom-Large-Precategory C)
is-retraction-hom-inv-iso-Large-Precategory = pr2 (pr2 (pr2 f))


## Examples

### The identity isomorphisms

For any object x : A, the identity morphism id_x : hom x x is an isomorphism from x to x since id_x ∘ id_x = id_x (it is its own inverse).

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 : Level} {X : obj-Large-Precategory C l1}
where

is-iso-id-hom-Large-Precategory :
is-iso-Large-Precategory C (id-hom-Large-Precategory C {X = X})
pr1 is-iso-id-hom-Large-Precategory = id-hom-Large-Precategory C
pr1 (pr2 is-iso-id-hom-Large-Precategory) =
left-unit-law-comp-hom-Large-Precategory C (id-hom-Large-Precategory C)
pr2 (pr2 is-iso-id-hom-Large-Precategory) =
left-unit-law-comp-hom-Large-Precategory C (id-hom-Large-Precategory C)

id-iso-Large-Precategory : iso-Large-Precategory C X X
pr1 id-iso-Large-Precategory = id-hom-Large-Precategory C
pr2 id-iso-Large-Precategory = is-iso-id-hom-Large-Precategory


## Properties

### Being an isomorphism is a proposition

Let f : hom x y and suppose g g' : hom y x are both two-sided inverses to f. It is enough to show that g = g' since the equalities are propositions (since the hom-types are sets). But we have the following chain of equalities: g = g ∘ id_y = g ∘ (f ∘ g') = (g ∘ f) ∘ g' = id_x ∘ g' = g'.

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
where

all-elements-equal-is-iso-Large-Precategory :
(f : hom-Large-Precategory C X Y)
(H K : is-iso-Large-Precategory C f) → H ＝ K
all-elements-equal-is-iso-Large-Precategory f (g , p , q) (g' , p' , q') =
eq-type-subtype
( λ g →
product-Prop
( Id-Prop
( hom-set-Large-Precategory C Y Y)
( comp-hom-Large-Precategory C f g)
( id-hom-Large-Precategory C))
( Id-Prop
( hom-set-Large-Precategory C X X)
( comp-hom-Large-Precategory C g f)
( id-hom-Large-Precategory C)))
( ( inv (right-unit-law-comp-hom-Large-Precategory C g)) ∙
( ap ( comp-hom-Large-Precategory C g) (inv p')) ∙
( inv (associative-comp-hom-Large-Precategory C g f g')) ∙
( ap ( comp-hom-Large-Precategory' C g') q) ∙
( left-unit-law-comp-hom-Large-Precategory C g'))

is-prop-is-iso-Large-Precategory :
(f : hom-Large-Precategory C X Y) →
is-prop (is-iso-Large-Precategory C f)
is-prop-is-iso-Large-Precategory f =
is-prop-all-elements-equal
( all-elements-equal-is-iso-Large-Precategory f)

is-iso-prop-Large-Precategory :
(f : hom-Large-Precategory C X Y) → Prop (β l1 l1 ⊔ β l2 l1 ⊔ β l2 l2)
pr1 (is-iso-prop-Large-Precategory f) =
is-iso-Large-Precategory C f
pr2 (is-iso-prop-Large-Precategory f) =
is-prop-is-iso-Large-Precategory f


### Equality of isomorphism is equality of their underlying morphisms

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
where

eq-iso-eq-hom-Large-Precategory :
(f g : iso-Large-Precategory C X Y) →
hom-iso-Large-Precategory C f ＝ hom-iso-Large-Precategory C g → f ＝ g
eq-iso-eq-hom-Large-Precategory f g =
eq-type-subtype (is-iso-prop-Large-Precategory C)


### The type of isomorphisms form a set

The type of isomorphisms between objects x y : A is a subtype of the set hom x y since being an isomorphism is a proposition.

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
where

is-set-iso-Large-Precategory : is-set (iso-Large-Precategory C X Y)
is-set-iso-Large-Precategory =
is-set-type-subtype
( is-iso-prop-Large-Precategory C)
( is-set-hom-Large-Precategory C X Y)

iso-set-Large-Precategory : Set (β l1 l1 ⊔ β l1 l2 ⊔ β l2 l1 ⊔ β l2 l2)
pr1 iso-set-Large-Precategory = iso-Large-Precategory C X Y
pr2 iso-set-Large-Precategory = is-set-iso-Large-Precategory


### Equalities induce isomorphisms

An equality between objects X Y : A gives rise to an isomorphism between them. This is because, by the J-rule, it is enough to construct an isomorphism given refl : X ＝ X, from X to itself. We take the identity morphism as such an isomorphism.

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 : Level}
where

iso-eq-Large-Precategory :
(X Y : obj-Large-Precategory C l1) → X ＝ Y → iso-Large-Precategory C X Y
pr1 (iso-eq-Large-Precategory X Y p) = hom-eq-Large-Precategory C X Y p
pr2 (iso-eq-Large-Precategory X .X refl) = is-iso-id-hom-Large-Precategory C

compute-iso-eq-Large-Precategory :
(X Y : obj-Large-Precategory C l1) →
iso-eq-Precategory (precategory-Large-Precategory C l1) X Y ~
iso-eq-Large-Precategory X Y
compute-iso-eq-Large-Precategory X Y p =
eq-iso-eq-hom-Large-Precategory C
( iso-eq-Precategory (precategory-Large-Precategory C l1) X Y p)
( iso-eq-Large-Precategory X Y p)
( compute-hom-eq-Large-Precategory C X Y p)


### Isomorphisms are closed under composition

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1}
{Y : obj-Large-Precategory C l2}
{Z : obj-Large-Precategory C l3}
{g : hom-Large-Precategory C Y Z}
{f : hom-Large-Precategory C X Y}
where

hom-comp-is-iso-Large-Precategory :
is-iso-Large-Precategory C g →
is-iso-Large-Precategory C f →
hom-Large-Precategory C Z X
hom-comp-is-iso-Large-Precategory q p =
comp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C f p)
( hom-inv-is-iso-Large-Precategory C g q)

is-section-comp-is-iso-Large-Precategory :
(q : is-iso-Large-Precategory C g)
(p : is-iso-Large-Precategory C f) →
comp-hom-Large-Precategory C
( comp-hom-Large-Precategory C g f)
( hom-comp-is-iso-Large-Precategory q p) ＝
id-hom-Large-Precategory C
is-section-comp-is-iso-Large-Precategory q p =
( associative-comp-hom-Large-Precategory C g f _) ∙
( ap
( comp-hom-Large-Precategory C g)
( ( inv
( associative-comp-hom-Large-Precategory C f
( hom-inv-is-iso-Large-Precategory C f p)
( hom-inv-is-iso-Large-Precategory C g q))) ∙
( ap
( λ h → comp-hom-Large-Precategory C h _)
( is-section-hom-inv-is-iso-Large-Precategory C f p)) ∙
( left-unit-law-comp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C g q)))) ∙
( is-section-hom-inv-is-iso-Large-Precategory C g q)

is-retraction-comp-is-iso-Large-Precategory :
(q : is-iso-Large-Precategory C g)
(p : is-iso-Large-Precategory C f) →
comp-hom-Large-Precategory C
( hom-comp-is-iso-Large-Precategory q p)
( comp-hom-Large-Precategory C g f) ＝
id-hom-Large-Precategory C
is-retraction-comp-is-iso-Large-Precategory q p =
( associative-comp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C f p)
( hom-inv-is-iso-Large-Precategory C g q)
( comp-hom-Large-Precategory C g f)) ∙
( ap
( comp-hom-Large-Precategory
( C)
( hom-inv-is-iso-Large-Precategory C f p))
( ( inv
( associative-comp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C g q)
( g)
( f))) ∙
( ap
( λ h → comp-hom-Large-Precategory C h f)
( is-retraction-hom-inv-is-iso-Large-Precategory C g q)) ∙
( left-unit-law-comp-hom-Large-Precategory C f))) ∙
( is-retraction-hom-inv-is-iso-Large-Precategory C f p)

is-iso-comp-is-iso-Large-Precategory :
is-iso-Large-Precategory C g → is-iso-Large-Precategory C f →
is-iso-Large-Precategory C (comp-hom-Large-Precategory C g f)
pr1 (is-iso-comp-is-iso-Large-Precategory q p) =
hom-comp-is-iso-Large-Precategory q p
pr1 (pr2 (is-iso-comp-is-iso-Large-Precategory q p)) =
is-section-comp-is-iso-Large-Precategory q p
pr2 (pr2 (is-iso-comp-is-iso-Large-Precategory q p)) =
is-retraction-comp-is-iso-Large-Precategory q p


### Composition of isomorphisms

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1}
{Y : obj-Large-Precategory C l2}
{Z : obj-Large-Precategory C l3}
(g : iso-Large-Precategory C Y Z)
(f : iso-Large-Precategory C X Y)
where

hom-comp-iso-Large-Precategory :
hom-Large-Precategory C X Z
hom-comp-iso-Large-Precategory =
comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C g)
( hom-iso-Large-Precategory C f)

is-iso-comp-iso-Large-Precategory :
is-iso-Large-Precategory C hom-comp-iso-Large-Precategory
is-iso-comp-iso-Large-Precategory =
is-iso-comp-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C g)
( is-iso-iso-Large-Precategory C f)

comp-iso-Large-Precategory :
iso-Large-Precategory C X Z
pr1 comp-iso-Large-Precategory = hom-comp-iso-Large-Precategory
pr2 comp-iso-Large-Precategory = is-iso-comp-iso-Large-Precategory

hom-inv-comp-iso-Large-Precategory :
hom-Large-Precategory C Z X
hom-inv-comp-iso-Large-Precategory =
hom-inv-iso-Large-Precategory C comp-iso-Large-Precategory

is-section-inv-comp-iso-Large-Precategory :
comp-hom-Large-Precategory C
( hom-comp-iso-Large-Precategory)
( hom-inv-comp-iso-Large-Precategory) ＝
id-hom-Large-Precategory C
is-section-inv-comp-iso-Large-Precategory =
is-section-hom-inv-iso-Large-Precategory C comp-iso-Large-Precategory

is-retraction-inv-comp-iso-Large-Precategory :
comp-hom-Large-Precategory C
( hom-inv-comp-iso-Large-Precategory)
( hom-comp-iso-Large-Precategory) ＝
id-hom-Large-Precategory C
is-retraction-inv-comp-iso-Large-Precategory =
is-retraction-hom-inv-iso-Large-Precategory C comp-iso-Large-Precategory


### Inverses of isomorphisms are isomorphisms

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
{f : hom-Large-Precategory C X Y}
where

is-iso-inv-is-iso-Large-Precategory :
(p : is-iso-Large-Precategory C f) →
is-iso-Large-Precategory C (hom-inv-iso-Large-Precategory C (f , p))
pr1 (is-iso-inv-is-iso-Large-Precategory p) = f
pr1 (pr2 (is-iso-inv-is-iso-Large-Precategory p)) =
is-retraction-hom-inv-is-iso-Large-Precategory C f p
pr2 (pr2 (is-iso-inv-is-iso-Large-Precategory p)) =
is-section-hom-inv-is-iso-Large-Precategory C f p


### Inverses of isomorphisms

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
where

inv-iso-Large-Precategory :
iso-Large-Precategory C X Y → iso-Large-Precategory C Y X
pr1 (inv-iso-Large-Precategory f) = hom-inv-iso-Large-Precategory C f
pr2 (inv-iso-Large-Precategory f) =
is-iso-inv-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C f)


### Composition of isomorphisms satisfies the unit laws

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : iso-Large-Precategory C X Y)
where

left-unit-law-comp-iso-Large-Precategory :
comp-iso-Large-Precategory C (id-iso-Large-Precategory C) f ＝ f
left-unit-law-comp-iso-Large-Precategory =
eq-iso-eq-hom-Large-Precategory C
( comp-iso-Large-Precategory C (id-iso-Large-Precategory C) f)
( f)
( left-unit-law-comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C f))

right-unit-law-comp-iso-Large-Precategory :
comp-iso-Large-Precategory C f (id-iso-Large-Precategory C) ＝ f
right-unit-law-comp-iso-Large-Precategory =
eq-iso-eq-hom-Large-Precategory C
( comp-iso-Large-Precategory C f (id-iso-Large-Precategory C))
( f)
( right-unit-law-comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C f))


### Composition of isomorphisms is associative

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 l4 : Level}
{X : obj-Large-Precategory C l1}
{Y : obj-Large-Precategory C l2}
{Z : obj-Large-Precategory C l3}
{W : obj-Large-Precategory C l4}
(h : iso-Large-Precategory C Z W)
(g : iso-Large-Precategory C Y Z)
(f : iso-Large-Precategory C X Y)
where

associative-comp-iso-Large-Precategory :
comp-iso-Large-Precategory C (comp-iso-Large-Precategory C h g) f ＝
comp-iso-Large-Precategory C h (comp-iso-Large-Precategory C g f)
associative-comp-iso-Large-Precategory =
eq-iso-eq-hom-Large-Precategory C
( comp-iso-Large-Precategory C (comp-iso-Large-Precategory C h g) f)
( comp-iso-Large-Precategory C h (comp-iso-Large-Precategory C g f))
( associative-comp-hom-Large-Precategory C
( hom-iso-Large-Precategory C h)
( hom-iso-Large-Precategory C g)
( hom-iso-Large-Precategory C f))


### Composition of isomorphisms satisfies inverse laws

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : iso-Large-Precategory C X Y)
where

left-inverse-law-comp-iso-Large-Precategory :
comp-iso-Large-Precategory C (inv-iso-Large-Precategory C f) f ＝
id-iso-Large-Precategory C
left-inverse-law-comp-iso-Large-Precategory =
eq-iso-eq-hom-Large-Precategory C
( comp-iso-Large-Precategory C (inv-iso-Large-Precategory C f) f)
( id-iso-Large-Precategory C)
( is-retraction-hom-inv-iso-Large-Precategory C f)

right-inverse-law-comp-iso-Large-Precategory :
comp-iso-Large-Precategory C f (inv-iso-Large-Precategory C f) ＝
id-iso-Large-Precategory C
right-inverse-law-comp-iso-Large-Precategory =
eq-iso-eq-hom-Large-Precategory C
( comp-iso-Large-Precategory C f (inv-iso-Large-Precategory C f))
( id-iso-Large-Precategory C)
( is-section-hom-inv-iso-Large-Precategory C f)


### A morphism f is an isomorphism if and only if precomposition by f is an equivalence

Proof: If f is an isomorphism with inverse f⁻¹, then precomposing with f⁻¹ is an inverse of precomposing with f. The only interesting direction is therefore the converse.

Suppose that precomposing with f is an equivalence, for any object Z. Then

  - ∘ f : hom Y X → hom X X


is an equivalence. In particular, there is a unique morphism g : Y → X such that g ∘ f ＝ id. Thus we have a retraction of f. To see that g is also a section, note that the map

  - ∘ f : hom Y Y → hom X Y


is an equivalence. In particular, it is injective. Therefore it suffices to show that (f ∘ g) ∘ f ＝ id ∘ f. To see this, we calculate

  (f ∘ g) ∘ f ＝ f ∘ (g ∘ f) ＝ f ∘ id ＝ f ＝ id ∘ f.


This completes the proof.

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
{f : hom-Large-Precategory C X Y}
(H :
{l3 : Level} (Z : obj-Large-Precategory C l3) →
is-equiv (precomp-hom-Large-Precategory C f Z))
where

hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory :
hom-Large-Precategory C Y X
hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory =
map-inv-is-equiv (H X) (id-hom-Large-Precategory C)

is-retraction-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory :
comp-hom-Large-Precategory C
( hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory)
( f) ＝
id-hom-Large-Precategory C
is-retraction-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory =
is-section-map-inv-is-equiv (H X) (id-hom-Large-Precategory C)

is-section-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory :
comp-hom-Large-Precategory C
( f)
( hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory) ＝
id-hom-Large-Precategory C
is-section-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory =
is-injective-is-equiv
( H Y)
( ( associative-comp-hom-Large-Precategory C
( f)
( hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory)
( f)) ∙
( ap
( comp-hom-Large-Precategory C f)
( is-retraction-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory)) ∙
( right-unit-law-comp-hom-Large-Precategory C f) ∙
( inv (left-unit-law-comp-hom-Large-Precategory C f)))

is-iso-is-equiv-precomp-hom-Large-Precategory :
is-iso-Large-Precategory C f
pr1 is-iso-is-equiv-precomp-hom-Large-Precategory =
hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory
pr1 (pr2 is-iso-is-equiv-precomp-hom-Large-Precategory) =
is-section-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory
pr2 (pr2 is-iso-is-equiv-precomp-hom-Large-Precategory) =
is-retraction-hom-inv-is-iso-is-equiv-precomp-hom-Large-Precategory

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
{f : hom-Large-Precategory C X Y}
(is-iso-f : is-iso-Large-Precategory C f)
(Z : obj-Large-Precategory C l3)
where

map-inv-precomp-hom-is-iso-Large-Precategory :
hom-Large-Precategory C X Z → hom-Large-Precategory C Y Z
map-inv-precomp-hom-is-iso-Large-Precategory =
precomp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C f is-iso-f)
( Z)

is-section-map-inv-precomp-hom-is-iso-Large-Precategory :
is-section
( precomp-hom-Large-Precategory C f Z)
( map-inv-precomp-hom-is-iso-Large-Precategory)
is-section-map-inv-precomp-hom-is-iso-Large-Precategory g =
( associative-comp-hom-Large-Precategory C
( g)
( hom-inv-is-iso-Large-Precategory C f is-iso-f)
( f)) ∙
( ap
( comp-hom-Large-Precategory C g)
( is-retraction-hom-inv-is-iso-Large-Precategory C f is-iso-f)) ∙
( right-unit-law-comp-hom-Large-Precategory C g)

is-retraction-map-inv-precomp-hom-is-iso-Large-Precategory :
is-retraction
( precomp-hom-Large-Precategory C f Z)
( map-inv-precomp-hom-is-iso-Large-Precategory)
is-retraction-map-inv-precomp-hom-is-iso-Large-Precategory g =
( associative-comp-hom-Large-Precategory C
( g)
( f)
( hom-inv-is-iso-Large-Precategory C f is-iso-f)) ∙
( ap
( comp-hom-Large-Precategory C g)
( is-section-hom-inv-is-iso-Large-Precategory C f is-iso-f)) ∙
( right-unit-law-comp-hom-Large-Precategory C g)

is-equiv-precomp-hom-is-iso-Large-Precategory :
is-equiv (precomp-hom-Large-Precategory C f Z)
is-equiv-precomp-hom-is-iso-Large-Precategory =
is-equiv-is-invertible
( map-inv-precomp-hom-is-iso-Large-Precategory)
( is-section-map-inv-precomp-hom-is-iso-Large-Precategory)
( is-retraction-map-inv-precomp-hom-is-iso-Large-Precategory)

equiv-precomp-hom-is-iso-Large-Precategory :
hom-Large-Precategory C Y Z ≃ hom-Large-Precategory C X Z
pr1 equiv-precomp-hom-is-iso-Large-Precategory =
precomp-hom-Large-Precategory C f Z
pr2 equiv-precomp-hom-is-iso-Large-Precategory =
is-equiv-precomp-hom-is-iso-Large-Precategory

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : iso-Large-Precategory C X Y)
(Z : obj-Large-Precategory C l3)
where

is-equiv-precomp-hom-iso-Large-Precategory :
is-equiv (precomp-hom-Large-Precategory C (hom-iso-Large-Precategory C f) Z)
is-equiv-precomp-hom-iso-Large-Precategory =
is-equiv-precomp-hom-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C f)
( Z)

equiv-precomp-hom-iso-Large-Precategory :
hom-Large-Precategory C Y Z ≃ hom-Large-Precategory C X Z
equiv-precomp-hom-iso-Large-Precategory =
equiv-precomp-hom-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C f)
( Z)


### A morphism f is an isomorphism if and only if postcomposition by f is an equivalence

Proof: If f is an isomorphism with inverse f⁻¹, then postcomposing with f⁻¹ is an inverse of postcomposing with f. The only interesting direction is therefore the converse.

Suppose that postcomposing with f is an equivalence, for any object Z. Then

  f ∘ - : hom Y X → hom Y Y


is an equivalence. In particular, there is a unique morphism g : Y → X such that f ∘ g ＝ id. Thus we have a section of f. To see that g is also a retraction, note that the map

  f ∘ - : hom X X → hom X Y


is an equivalence. In particular, it is injective. Therefore it suffices to show that f ∘ (g ∘ f) ＝ f ∘ id. To see this, we calculate

  f ∘ (g ∘ f) ＝ (f ∘ g) ∘ f ＝ id ∘ f ＝ f ＝ f ∘ id.


This completes the proof.

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
{f : hom-Large-Precategory C X Y}
(H :
{l3 : Level} (Z : obj-Large-Precategory C l3) →
is-equiv (postcomp-hom-Large-Precategory C Z f))
where

hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory :
hom-Large-Precategory C Y X
hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory =
map-inv-is-equiv (H Y) (id-hom-Large-Precategory C)

is-section-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory :
comp-hom-Large-Precategory C
( f)
( hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory) ＝
id-hom-Large-Precategory C
is-section-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory =
is-section-map-inv-is-equiv (H Y) (id-hom-Large-Precategory C)

is-retraction-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory :
comp-hom-Large-Precategory C
( hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory)
( f) ＝
id-hom-Large-Precategory C
is-retraction-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory =
is-injective-is-equiv
( H X)
( ( inv
( associative-comp-hom-Large-Precategory C
( f)
( hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory)
( f))) ∙
( ap
( comp-hom-Large-Precategory' C f)
( is-section-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory)) ∙
( left-unit-law-comp-hom-Large-Precategory C f) ∙
( inv (right-unit-law-comp-hom-Large-Precategory C f)))

is-iso-is-equiv-postcomp-hom-Large-Precategory :
is-iso-Large-Precategory C f
pr1 is-iso-is-equiv-postcomp-hom-Large-Precategory =
hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory
pr1 (pr2 is-iso-is-equiv-postcomp-hom-Large-Precategory) =
is-section-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory
pr2 (pr2 is-iso-is-equiv-postcomp-hom-Large-Precategory) =
is-retraction-hom-inv-is-iso-is-equiv-postcomp-hom-Large-Precategory

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
{f : hom-Large-Precategory C X Y}
(is-iso-f : is-iso-Large-Precategory C f)
(Z : obj-Large-Precategory C l3)
where

map-inv-postcomp-hom-is-iso-Large-Precategory :
hom-Large-Precategory C Z Y → hom-Large-Precategory C Z X
map-inv-postcomp-hom-is-iso-Large-Precategory =
postcomp-hom-Large-Precategory C
( Z)
( hom-inv-is-iso-Large-Precategory C f is-iso-f)

is-section-map-inv-postcomp-hom-is-iso-Large-Precategory :
is-section
( postcomp-hom-Large-Precategory C Z f)
( map-inv-postcomp-hom-is-iso-Large-Precategory)
is-section-map-inv-postcomp-hom-is-iso-Large-Precategory g =
( inv
( associative-comp-hom-Large-Precategory C
( f)
( hom-inv-is-iso-Large-Precategory C f is-iso-f)
( g))) ∙
( ap
( comp-hom-Large-Precategory' C g)
( is-section-hom-inv-is-iso-Large-Precategory C f is-iso-f)) ∙
( left-unit-law-comp-hom-Large-Precategory C g)

is-retraction-map-inv-postcomp-hom-is-iso-Large-Precategory :
is-retraction
( postcomp-hom-Large-Precategory C Z f)
( map-inv-postcomp-hom-is-iso-Large-Precategory)
is-retraction-map-inv-postcomp-hom-is-iso-Large-Precategory g =
( inv
( associative-comp-hom-Large-Precategory C
( hom-inv-is-iso-Large-Precategory C f is-iso-f)
( f)
( g))) ∙
( ap
( comp-hom-Large-Precategory' C g)
( is-retraction-hom-inv-is-iso-Large-Precategory C f is-iso-f)) ∙
( left-unit-law-comp-hom-Large-Precategory C g)

is-equiv-postcomp-hom-is-iso-Large-Precategory :
is-equiv (postcomp-hom-Large-Precategory C Z f)
is-equiv-postcomp-hom-is-iso-Large-Precategory =
is-equiv-is-invertible
( map-inv-postcomp-hom-is-iso-Large-Precategory)
( is-section-map-inv-postcomp-hom-is-iso-Large-Precategory)
( is-retraction-map-inv-postcomp-hom-is-iso-Large-Precategory)

equiv-postcomp-hom-is-iso-Large-Precategory :
hom-Large-Precategory C Z X ≃ hom-Large-Precategory C Z Y
pr1 equiv-postcomp-hom-is-iso-Large-Precategory =
postcomp-hom-Large-Precategory C Z f
pr2 equiv-postcomp-hom-is-iso-Large-Precategory =
is-equiv-postcomp-hom-is-iso-Large-Precategory

module _
{α : Level → Level} {β : Level → Level → Level}
(C : Large-Precategory α β) {l1 l2 l3 : Level}
{X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2}
(f : iso-Large-Precategory C X Y)
(Z : obj-Large-Precategory C l3)
where

is-equiv-postcomp-hom-iso-Large-Precategory :
is-equiv
( postcomp-hom-Large-Precategory C Z (hom-iso-Large-Precategory C f))
is-equiv-postcomp-hom-iso-Large-Precategory =
is-equiv-postcomp-hom-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C f)
( Z)

equiv-postcomp-hom-iso-Large-Precategory :
hom-Large-Precategory C Z X ≃ hom-Large-Precategory C Z Y
equiv-postcomp-hom-iso-Large-Precategory =
equiv-postcomp-hom-is-iso-Large-Precategory C
( is-iso-iso-Large-Precategory C f)
( Z)