open import Cat.Diagram.Limit.Base
open import Cat.Morphism
open import Cat.Prelude hiding (J)

import Cat.Functor.Reasoning as Func
import Cat.Reasoning as Cat

module Cat.Functor.Conservative where

private variable
o h o₁ h₁ : Level
C D J : Precategory o h
open Precategory
open Functor


# Conservative functors🔗

We say a functor is conservative if it reflects isomorphisms. More concretely, if $f : A \to B$ is some morphism $\mathcal{C}$, and if $F(f)$ is an iso in $\mathcal{D}$, then $f$ must have already been an iso in $\mathcal{C}$!

is-conservative : Functor C D → Type _
is-conservative {C = C} {D = D} F =
∀ {A B} {f : C .Hom A B}
→ is-invertible D (F .F₁ f) → is-invertible C f


As a general fact, conservative functors reflect limits that they preserve (given those limits exist in the first place!).

The rough proof sketch is as follows: Let $K$ be some cone in $C$ such that $F(K)$ is a limit in $D$, and $L$ a limit in $C$ of the same diagram. By the universal property of $L$, there exists a map $\eta$ from the apex of $K$ to the apex of $L$ in $C$. Furthermore, as $F(K)$ is a limit in $D$, $F(\eta)$ becomes an isomorphism in $D$. However, $F$ is conservative, which implies that $\eta$ was an isomorphism in $C$ all along! This means that $K$ must be a limit in $C$ as well (see is-invertible→is-limitp).

module _ {F : Functor C D} (conservative : is-conservative F) where
private
open _=>_
module C = Cat C
module D = Cat D
module F = Func F

conservative-reflects-limits : ∀ {Dia : Functor J C}
→ (L : Limit Dia)
→ preserves-limit F Dia
→ reflects-limit F Dia
conservative-reflects-limits L-lim preservesa {K} {eps} lim =
is-invertible→is-limitp
{K = Limit.Ext L-lim} {epsy = Limit.cone L-lim} (Limit.has-limit L-lim)
(eps .η) (λ f → sym (eps .is-natural _ _ f) ∙ C.elimr (K .F-id)) refl
$conservative$ invert

where
module L-lim = Limit L-lim
module FL-lim = is-limit (preservesa L-lim.has-limit)
module lim = is-limit lim

uinv : D.Hom (F .F₀ L-lim.apex) (F .F₀ (K .F₀ tt))
uinv =
(lim.universal
(λ j → F .F₁ (L-lim.ψ j))
(λ f → sym (F .F-∘ _ _) ∙ ap (F .F₁) (L-lim.commutes f)))

invert : D.is-invertible (F .F₁ (L-lim.universal (eps .η) _))
invert =
D.make-invertible uinv
(FL-lim.unique₂ _ (λ j → FL-lim.commutes j)
(λ j → F.pulll (L-lim.factors _ _) ∙ lim.factors _ _)
(λ j → D.idr _))
(lim.unique₂ _ (λ j → lim.commutes j)
(λ j → D.pulll (lim.factors _ _) ∙ F.collapse (L-lim.factors _ _))
(λ j → D.idr _))