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Mathlib.CategoryTheory.Functor.ReflectsIso

Functors which reflect isomorphisms #

A functor F reflects isomorphisms if whenever F.map f is an isomorphism, f was too.

It is formalized as a Prop valued typeclass ReflectsIsomorphisms F.

Any fully faithful functor reflects isomorphisms.

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class CategoryTheory.ReflectsIsomorphisms {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) :
Prop

Define what it means for a functor F : C ⥤ D to reflect isomorphisms: for any morphism f : A ⟶ B, if F.map f is an isomorphism then f is as well. Note that we do not assume or require that F is faithful.

Instances
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    theorem CategoryTheory.isIso_of_reflects_iso {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {A : C} {B : C} (f : A ⟶ B) (F : CategoryTheory.Functor C D) [CategoryTheory.IsIso (F.toPrefunctor.map f)] [CategoryTheory.ReflectsIsomorphisms F] :
    CategoryTheory.IsIso f

    If F reflects isos and F.map f is an iso, then f is an iso.

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    instance CategoryTheory.reflectsIsomorphisms_of_full_and_faithful {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] (F : CategoryTheory.Functor C D) [CategoryTheory.Full F] [CategoryTheory.Faithful F] :
    CategoryTheory.ReflectsIsomorphisms F
    Equations
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    instance CategoryTheory.reflectsIsomorphisms_of_comp {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] {E : Type u₃} [CategoryTheory.Category.{v₃, u₃} E] (F : CategoryTheory.Functor C D) (G : CategoryTheory.Functor D E) [CategoryTheory.ReflectsIsomorphisms F] [CategoryTheory.ReflectsIsomorphisms G] :
    CategoryTheory.ReflectsIsomorphisms (CategoryTheory.Functor.comp F G)
    Equations
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    instance CategoryTheory.reflectsIsomorphisms_of_reflectsMonomorphisms_of_reflectsEpimorphisms {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] {D : Type u₂} [CategoryTheory.Category.{v₂, u₂} D] [CategoryTheory.Balanced C] (F : CategoryTheory.Functor C D) [CategoryTheory.Functor.ReflectsMonomorphisms F] [CategoryTheory.Functor.ReflectsEpimorphisms F] :
    CategoryTheory.ReflectsIsomorphisms F
    Equations