# Conservative functors between precategories

Content created by Fredrik Bakke.

Created on 2023-11-01.

module category-theory.conservative-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.iterated-dependent-product-types
open import foundation.propositions
open import foundation.universe-levels


## Idea

A functor F : C → D between precategories is conservative if every morphism that is mapped to an isomorphism in D is an isomorphism in C.

## Definitions

### The predicate on functors of being conservative

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

is-conservative-functor-Precategory : UU (l1 ⊔ l2 ⊔ l4)
is-conservative-functor-Precategory =
{x y : obj-Precategory C} (f : hom-Precategory C x y) →
is-iso-Precategory D (hom-functor-Precategory C D F f) →
is-iso-Precategory C f

is-prop-is-conservative-functor-Precategory :
is-prop is-conservative-functor-Precategory
is-prop-is-conservative-functor-Precategory =
is-prop-iterated-implicit-Π 2
( λ x y → is-prop-iterated-Π 2 (λ f _ → is-prop-is-iso-Precategory C f))

is-conservative-prop-functor-Precategory : Prop (l1 ⊔ l2 ⊔ l4)
pr1 is-conservative-prop-functor-Precategory =
is-conservative-functor-Precategory
pr2 is-conservative-prop-functor-Precategory =
is-prop-is-conservative-functor-Precategory


### The type of conservative functors

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

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

functor-conservative-functor-Precategory :
functor-Precategory C D
functor-conservative-functor-Precategory = pr1 F

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

obj-conservative-functor-Precategory :
obj-Precategory C → obj-Precategory D
obj-conservative-functor-Precategory =
obj-functor-Precategory C D
( functor-conservative-functor-Precategory)

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