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Mathlib.CategoryTheory.Functor.ReflectsIso.Basic

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.Functor.ReflectsIsomorphisms {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] (F : 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.

  • reflects {A B : C} (f : A ⟶ B) [IsIso (F.map f)] : IsIso f

    For any f, if F.map f is an iso, then so was f.

Instances
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    theorem CategoryTheory.isIso_of_reflects_iso {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {A B : C} (f : A ⟶ B) (F : Functor C D) [IsIso (F.map f)] [F.ReflectsIsomorphisms] :
    IsIso f

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

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    theorem CategoryTheory.isIso_iff_of_reflects_iso {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {A B : C} (f : A ⟶ B) (F : Functor C D) [F.ReflectsIsomorphisms] :
    IsIso (F.map f) ↔ IsIso f
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    theorem CategoryTheory.Functor.FullyFaithful.reflectsIsomorphisms {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {F : Functor C D} (hF : F.FullyFaithful) :
    F.ReflectsIsomorphisms
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    @[instance 100]
    instance CategoryTheory.reflectsIsomorphisms_of_full_and_faithful {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] (F : Functor C D) [F.Full] [F.Faithful] :
    F.ReflectsIsomorphisms
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    instance CategoryTheory.reflectsIsomorphisms_comp {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {E : Type u_3} [Category.{v_3, u_3} E] (F : Functor C D) (G : Functor D E) [F.ReflectsIsomorphisms] [G.ReflectsIsomorphisms] :
    (F.comp G).ReflectsIsomorphisms
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    theorem CategoryTheory.reflectsIsomorphisms_of_comp {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {E : Type u_3} [Category.{v_3, u_3} E] (F : Functor C D) (G : Functor D E) [(F.comp G).ReflectsIsomorphisms] :
    F.ReflectsIsomorphisms
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    instance CategoryTheory.instReflectsIsomorphismsFunctorObjWhiskeringRight {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {E : Type u_3} [Category.{v_3, u_3} E] (F : Functor D E) [F.ReflectsIsomorphisms] :
    ((Functor.whiskeringRight C D E).obj F).ReflectsIsomorphisms
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    theorem CategoryTheory.reflectsIsomorphisms_of_iso {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {F G : Functor C D} (α : F ≅ G) [F.ReflectsIsomorphisms] :
    G.ReflectsIsomorphisms
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    theorem CategoryTheory.reflectsIsomorphisms_iso_iff {C : Type u_1} [Category.{v_1, u_1} C] {D : Type u_2} [Category.{v_2, u_2} D] {F G : Functor C D} (α : F ≅ G) :
    F.ReflectsIsomorphisms ↔ G.ReflectsIsomorphisms