# Essentially injective functors between precategories

Content created by Fredrik Bakke.

Created on 2023-11-01.

module category-theory.essentially-injective-functors-precategories where

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

open import foundation.dependent-pair-types
open import foundation.universe-levels


## Idea

A functor F : C → D between precategories is essentially injective if every pair of objects that are mapped to isomorphic objects in D are isomorphic in C.

## Definitions

### The type of proofs of being essentially injective

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

is-essentially-injective-functor-Precategory : UU (l1 ⊔ l2 ⊔ l4)
is-essentially-injective-functor-Precategory =
(x y : obj-Precategory C) →
iso-Precategory D
( obj-functor-Precategory C D F x)
( obj-functor-Precategory C D F y) →
iso-Precategory C x y


### The type of essentially injective functors

essentially-injective-functor-Precategory :
{l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (D : Precategory l3 l4) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
essentially-injective-functor-Precategory C D =
Σ ( functor-Precategory C D)
( is-essentially-injective-functor-Precategory C D)

module _
{l1 l2 l3 l4 : Level}
(C : Precategory l1 l2) (D : Precategory l3 l4)
(F : essentially-injective-functor-Precategory C D)
where

functor-essentially-injective-functor-Precategory :
functor-Precategory C D
functor-essentially-injective-functor-Precategory = pr1 F

is-essentially-injective-essentially-injective-functor-Precategory :
is-essentially-injective-functor-Precategory C D
( functor-essentially-injective-functor-Precategory)
is-essentially-injective-essentially-injective-functor-Precategory = pr2 F

obj-essentially-injective-functor-Precategory :
obj-Precategory C → obj-Precategory D
obj-essentially-injective-functor-Precategory =
obj-functor-Precategory C D
( functor-essentially-injective-functor-Precategory)

hom-essentially-injective-functor-Precategory :
{x y : obj-Precategory C} →
hom-Precategory C x y →
hom-Precategory D
( obj-essentially-injective-functor-Precategory x)
( obj-essentially-injective-functor-Precategory y)
hom-essentially-injective-functor-Precategory =
hom-functor-Precategory C D
( functor-essentially-injective-functor-Precategory)