Isomorphisms

This file defines isomorphisms between objects of a category.

Main definitions

• structure Iso : a bundled isomorphism between two objects of a category;
• class IsIso : an unbundled version of Iso; note that IsIso f is a Prop, and only asserts the existence of an inverse. Of course, this inverse is unique, so it doesn't cost us much to use choice to retrieve it.
• inv f, for the inverse of a morphism with [IsIso f]
• asIso : convert from IsIso to Iso (noncomputable);
• of_iso : convert from Iso to IsIso;
• standard operations on isomorphisms (composition, inverse etc)

Notation

• X ≅ Y : same as Iso X Y;
• α ≪≫ β : composition of two isomorphisms; it is called Iso.trans

Tags

category, category theory, isomorphism

structure CategoryTheory.Iso {C : Type u} [Category.{v, u} C] (X Y : C) : Type v

An isomorphism (a.k.a. an invertible morphism) between two objects of a category. The inverse morphism is bundled.

See also CategoryTheory.Core for the category with the same objects and isomorphisms playing the role of morphisms.

• hom : X ⟶ Y

  The forward direction of an isomorphism.

• inv : Y ⟶ X

  The backwards direction of an isomorphism.

• hom_inv_id : CategoryStruct.comp self.hom self.inv = CategoryStruct.id X

  Composition of the two directions of an isomorphism is the identity on the source.

• inv_hom_id : CategoryStruct.comp self.inv self.hom = CategoryStruct.id Y

  Composition of the two directions of an isomorphism in reverse order is the identity on the target.

@[simp]

theorem CategoryTheory.Iso.hom_inv_id_assoc {C : Type u} [Category.{v, u} C] {X Y : C} (self : X ≅ Y) {Z : C} (h : X ⟶ Z) : CategoryStruct.comp self.hom (CategoryStruct.comp self.inv h) = h

Composition of the two directions of an isomorphism is the identity on the source.

@[simp]

theorem CategoryTheory.Iso.inv_hom_id_assoc {C : Type u} [Category.{v, u} C] {X Y : C} (self : X ≅ Y) {Z : C} (h : Y ⟶ Z) : CategoryStruct.comp self.inv (CategoryStruct.comp self.hom h) = h

Composition of the two directions of an isomorphism in reverse order is the identity on the target.

def CategoryTheory.«term_≅_» : Lean.TrailingParserDescr

Notation for an isomorphism in a category.

theorem CategoryTheory.Iso.ext {C : Type u} [Category.{v, u} C] {X Y : C} ⦃α β : X ≅ Y⦄ (w : α.hom = β.hom) : α = β

theorem CategoryTheory.Iso.ext_iff {C : Type u} [Category.{v, u} C] {X Y : C} {α β : X ≅ Y} : α = β ↔ α.hom = β.hom

theorem CategoryTheory.Iso.ext_inv {C : Type u} [Category.{v, u} C] {X Y : C} ⦃α β : X ≅ Y⦄ (w : α.inv = β.inv) : α = β

def CategoryTheory.Iso.symm {C : Type u} [Category.{v, u} C] {X Y : C} (I : X ≅ Y) : Y ≅ X

Inverse isomorphism.

@[simp]

theorem CategoryTheory.Iso.symm_hom {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α.symm.hom = α.inv

@[simp]

theorem CategoryTheory.Iso.symm_inv {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α.symm.inv = α.hom

@[simp]

theorem CategoryTheory.Iso.symm_mk {C : Type u} [Category.{v, u} C] {X Y : C} (hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id : CategoryStruct.comp hom inv = CategoryStruct.id X) (inv_hom_id : CategoryStruct.comp inv hom = CategoryStruct.id Y) : { hom := hom, inv := inv, hom_inv_id := hom_inv_id, inv_hom_id := inv_hom_id }.symm = { hom := inv, inv := hom, hom_inv_id := inv_hom_id, inv_hom_id := hom_inv_id }

@[simp]

theorem CategoryTheory.Iso.symm_symm_eq {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α.symm.symm = α

theorem CategoryTheory.Iso.symm_bijective {C : Type u} [Category.{v, u} C] {X Y : C} : Function.Bijective symm

@[simp]

theorem CategoryTheory.Iso.symm_eq_iff {C : Type u} [Category.{v, u} C] {X Y : C} {α β : X ≅ Y} : α.symm = β.symm ↔ α = β

theorem CategoryTheory.Iso.nonempty_iso_symm {C : Type u} [Category.{v, u} C] (X Y : C) : Nonempty (X ≅ Y) ↔ Nonempty (Y ≅ X)

def CategoryTheory.Iso.refl {C : Type u} [Category.{v, u} C] (X : C) : X ≅ X

Identity isomorphism.

@[simp]

theorem CategoryTheory.Iso.refl_hom {C : Type u} [Category.{v, u} C] (X : C) : (refl X).hom = CategoryStruct.id X

@[simp]

theorem CategoryTheory.Iso.refl_inv {C : Type u} [Category.{v, u} C] (X : C) : (refl X).inv = CategoryStruct.id X

@[implicit_reducible]

instance CategoryTheory.Iso.instInhabited {C : Type u} [Category.{v, u} C] {X : C} : Inhabited (X ≅ X)

theorem CategoryTheory.Iso.nonempty_iso_refl {C : Type u} [Category.{v, u} C] (X : C) : Nonempty (X ≅ X)

@[simp]

theorem CategoryTheory.Iso.refl_symm {C : Type u} [Category.{v, u} C] (X : C) : (refl X).symm = refl X

def CategoryTheory.Iso.trans {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : Y ≅ Z) : X ≅ Z

Composition of two isomorphisms

@[simp]

theorem CategoryTheory.Iso.trans_inv {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : Y ≅ Z) : (α ≪≫ β).inv = CategoryStruct.comp β.inv α.inv

@[simp]

theorem CategoryTheory.Iso.trans_hom {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : Y ≅ Z) : (α ≪≫ β).hom = CategoryStruct.comp α.hom β.hom

@[implicit_reducible]

instance CategoryTheory.Iso.instTransIso {C : Type u} [Category.{v, u} C] : Trans (fun (x1 x2 : C) => x1 ≅ x2) (fun (x1 x2 : C) => x1 ≅ x2) fun (x1 x2 : C) => x1 ≅ x2

@[simp]

theorem CategoryTheory.Iso.trans_def {C : Type u} [Category.{v, u} C] {a✝ b✝ c✝ : C} (α : a✝ ≅ b✝) (β : b✝ ≅ c✝) : Trans.trans α β = α ≪≫ β

def CategoryTheory.Iso.«term_≪≫_» : Lean.TrailingParserDescr

Notation for composition of isomorphisms.

@[simp]

theorem CategoryTheory.Iso.trans_mk {C : Type u} [Category.{v, u} C] {X Y Z : C} (hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id : CategoryStruct.comp hom inv = CategoryStruct.id X) (inv_hom_id : CategoryStruct.comp inv hom = CategoryStruct.id Y) (hom' : Y ⟶ Z) (inv' : Z ⟶ Y) (hom_inv_id' : CategoryStruct.comp hom' inv' = CategoryStruct.id Y) (inv_hom_id' : CategoryStruct.comp inv' hom' = CategoryStruct.id Z) (hom_inv_id'' : CategoryStruct.comp (CategoryStruct.comp hom hom') (CategoryStruct.comp inv' inv) = CategoryStruct.id X) (inv_hom_id'' : CategoryStruct.comp (CategoryStruct.comp inv' inv) (CategoryStruct.comp hom hom') = CategoryStruct.id Z) : { hom := hom, inv := inv, hom_inv_id := hom_inv_id, inv_hom_id := inv_hom_id } ≪≫ { hom := hom', inv := inv', hom_inv_id := hom_inv_id', inv_hom_id := inv_hom_id' } = { hom := CategoryStruct.comp hom hom', inv := CategoryStruct.comp inv' inv, hom_inv_id := hom_inv_id'', inv_hom_id := inv_hom_id'' }

@[simp]

theorem CategoryTheory.Iso.trans_symm {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : Y ≅ Z) : (α ≪≫ β).symm = β.symm ≪≫ α.symm

@[simp]

theorem CategoryTheory.Iso.trans_assoc {C : Type u} [Category.{v, u} C] {X Y Z Z' : C} (α : X ≅ Y) (β : Y ≅ Z) (γ : Z ≅ Z') : (α ≪≫ β) ≪≫ γ = α ≪≫ β ≪≫ γ

@[simp]

theorem CategoryTheory.Iso.refl_trans {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : refl X ≪≫ α = α

@[simp]

theorem CategoryTheory.Iso.trans_refl {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α ≪≫ refl Y = α

@[simp]

theorem CategoryTheory.Iso.symm_self_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α.symm ≪≫ α = refl Y

@[simp]

theorem CategoryTheory.Iso.self_symm_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) : α ≪≫ α.symm = refl X

@[simp]

theorem CategoryTheory.Iso.symm_self_id_assoc {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : Y ≅ Z) : α.symm ≪≫ α ≪≫ β = β

@[simp]

theorem CategoryTheory.Iso.self_symm_id_assoc {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) (β : X ≅ Z) : α ≪≫ α.symm ≪≫ β = β

theorem CategoryTheory.Iso.inv_comp_eq {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : CategoryStruct.comp α.inv f = g ↔ f = CategoryStruct.comp α.hom g

theorem CategoryTheory.Iso.eq_inv_comp {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : g = CategoryStruct.comp α.inv f ↔ CategoryStruct.comp α.hom g = f

theorem CategoryTheory.Iso.comp_inv_eq {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : CategoryStruct.comp f α.inv = g ↔ f = CategoryStruct.comp g α.hom

theorem CategoryTheory.Iso.eq_comp_inv {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : g = CategoryStruct.comp f α.inv ↔ CategoryStruct.comp g α.hom = f

theorem CategoryTheory.Iso.inv_eq_inv {C : Type u} [Category.{v, u} C] {X Y : C} (f g : X ≅ Y) : f.inv = g.inv ↔ f.hom = g.hom

theorem CategoryTheory.Iso.hom_comp_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {f : Y ⟶ X} : CategoryStruct.comp α.hom f = CategoryStruct.id X ↔ f = α.inv

theorem CategoryTheory.Iso.comp_inv_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {f : X ⟶ Y} : CategoryStruct.comp f α.inv = CategoryStruct.id X ↔ f = α.hom

theorem CategoryTheory.Iso.comp_hom_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {f : Y ⟶ X} : CategoryStruct.comp f α.hom = CategoryStruct.id Y ↔ f = α.inv

theorem CategoryTheory.Iso.inv_comp_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {f : X ⟶ Y} : CategoryStruct.comp α.inv f = CategoryStruct.id Y ↔ f = α.hom

theorem CategoryTheory.Iso.hom_eq_inv {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) (β : Y ≅ X) : α.hom = β.inv ↔ β.hom = α.inv

theorem CategoryTheory.Iso.inv_eq_hom {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) (β : Y ≅ X) : α.inv = β.hom ↔ β.inv = α.hom

def CategoryTheory.Iso.homToEquiv {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} : (Z ⟶ X) ≃ (Z ⟶ Y)

The bijection (Z ⟶ X) ≃ (Z ⟶ Y) induced by α : X ≅ Y.

def CategoryTheory.Iso.homFromEquiv {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} : (X ⟶ Z) ≃ (Y ⟶ Z)

The bijection (X ⟶ Z) ≃ (Y ⟶ Z) induced by α : X ≅ Y.

@[simp]

theorem CategoryTheory.Iso.homFromEquiv_symm_apply {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} (g : Y ⟶ Z) : α.homFromEquiv.symm g = CategoryStruct.comp α.hom g

@[simp]

theorem CategoryTheory.Iso.homFromEquiv_apply {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} (f : X ⟶ Z) : α.homFromEquiv f = CategoryStruct.comp α.inv f

@[simp]

theorem CategoryTheory.Iso.homToEquiv_apply {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} (f : Z ⟶ X) : α.homToEquiv f = CategoryStruct.comp f α.hom

@[simp]

theorem CategoryTheory.Iso.homToEquiv_symm_apply {C : Type u} [Category.{v, u} C] {X Y : C} (α : X ≅ Y) {Z : C} (g : Z ⟶ Y) : α.homToEquiv.symm g = CategoryStruct.comp g α.inv

class CategoryTheory.IsIso {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) : Prop

The IsIso typeclass expresses that a morphism is invertible.

Given a morphism f with IsIso f, one can view f as an isomorphism via asIso f and get the inverse using inv f.

• out : ∃ (inv : Y ⟶ X), CategoryStruct.comp f inv = CategoryStruct.id X ∧ CategoryStruct.comp inv f = CategoryStruct.id Y

  The existence of an inverse morphism.

theorem CategoryTheory.IsIso.mk' {C : Type u} [Category.{v, u} C] {X Y : C} {f : Y ⟶ X} (out : ∃ (inv : X ⟶ Y), CategoryStruct.comp inv f = CategoryStruct.id X ∧ CategoryStruct.comp f inv = CategoryStruct.id Y) : IsIso f

IsIso.mk' is the dual of IsIso.mk, which we need for to_dual. Please avoid using this directly.

noncomputable def CategoryTheory.inv {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [I : IsIso f] : Y ⟶ X

The inverse of a morphism f when we have [IsIso f].

@[simp]

theorem CategoryTheory.IsIso.hom_inv_id {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [I : IsIso f] : CategoryStruct.comp f (inv f) = CategoryStruct.id X

@[simp]

theorem CategoryTheory.IsIso.inv_hom_id {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [I : IsIso f] : CategoryStruct.comp (inv f) f = CategoryStruct.id Y

@[simp]

theorem CategoryTheory.IsIso.hom_inv_id_assoc {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [I : IsIso f] {Z : C} (h : X ⟶ Z) : CategoryStruct.comp f (CategoryStruct.comp (inv f) h) = h

@[simp]

theorem CategoryTheory.IsIso.inv_hom_id_assoc {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [I : IsIso f] {Z : C} (h : Y ⟶ Z) : CategoryStruct.comp (inv f) (CategoryStruct.comp f h) = h

instance CategoryTheory.Iso.isIso_hom {C : Type u} [Category.{v, u} C] {X Y : C} (e : X ≅ Y) : IsIso e.hom

instance CategoryTheory.Iso.isIso_inv {C : Type u} [Category.{v, u} C] {X Y : C} (e : X ≅ Y) : IsIso e.inv

noncomputable def CategoryTheory.asIso {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [IsIso f] : X ≅ Y

Reinterpret a morphism f : X ⟶ Y with an IsIso f instance as X ≅ Y.

noncomputable def CategoryTheory.asIso' {C : Type u} [Category.{v, u} C] {X Y : C} (f : Y ⟶ X) [IsIso f] : X ≅ Y

Reinterpret a morphism f : X ⟶ Y with an IsIso f instance as Y ≅ X.

@[simp]

theorem CategoryTheory.asIso_hom {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [IsIso f] : (asIso f).hom = f

@[simp]

theorem CategoryTheory.asIso'_hom {C : Type u} [Category.{v, u} C] {X Y : C} (f : Y ⟶ X) [IsIso f] : (asIso' f).inv = f

@[simp]

theorem CategoryTheory.asIso_inv {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [IsIso f] : (asIso f).inv = inv f

@[simp]

theorem CategoryTheory.asIso'_inv {C : Type u} [Category.{v, u} C] {X Y : C} (f : Y ⟶ X) [IsIso f] : (asIso' f).hom = inv f

@[instance 100]

instance CategoryTheory.IsIso.epi_of_iso {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ⟶ Y) [IsIso f] : Epi f

@[instance 100]

instance CategoryTheory.IsIso.mono_of_iso {C : Type u} [Category.{v, u} C] {X Y : C} (f : Y ⟶ X) [IsIso f] : Mono f

theorem CategoryTheory.IsIso.inv_eq_of_hom_inv_id {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ⟶ Y} [IsIso f] {g : Y ⟶ X} (hom_inv_id : CategoryStruct.comp f g = CategoryStruct.id X) : inv f = g

theorem CategoryTheory.IsIso.inv_eq_of_inv_hom_id {C : Type u} [Category.{v, u} C] {X Y : C} {f : Y ⟶ X} [IsIso f] {g : X ⟶ Y} (hom_inv_id : CategoryStruct.comp g f = CategoryStruct.id X) : inv f = g

theorem CategoryTheory.IsIso.eq_inv_of_hom_inv_id {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ⟶ Y} [IsIso f] {g : Y ⟶ X} (hom_inv_id : CategoryStruct.comp f g = CategoryStruct.id X) : g = inv f

theorem CategoryTheory.IsIso.eq_inv_of_inv_hom_id {C : Type u} [Category.{v, u} C] {X Y : C} {f : Y ⟶ X} [IsIso f] {g : X ⟶ Y} (hom_inv_id : CategoryStruct.comp g f = CategoryStruct.id X) : g = inv f

instance CategoryTheory.IsIso.id {C : Type u} [Category.{v, u} C] (X : C) : IsIso (CategoryStruct.id X)

instance CategoryTheory.IsIso.inv_isIso {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ⟶ Y} [IsIso f] : IsIso (inv f)

@[instance 900]

instance CategoryTheory.IsIso.comp_isIso {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} [IsIso f] [IsIso h] : IsIso (CategoryStruct.comp f h)

theorem CategoryTheory.IsIso.comp_isIso' {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} : IsIso f → IsIso h → IsIso (CategoryStruct.comp f h)

The composition of isomorphisms is an isomorphism. Here the arguments of type IsIso are explicit, to make this easier to use with the refine tactic, for instance.

@[simp]

theorem CategoryTheory.IsIso.inv_id {C : Type u} [Category.{v, u} C] {X : C} : inv (CategoryStruct.id X) = CategoryStruct.id X

@[simp]

theorem CategoryTheory.IsIso.inv_comp {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} [IsIso f] [IsIso h] : inv (CategoryStruct.comp f h) = CategoryStruct.comp (inv h) (inv f)

theorem CategoryTheory.IsIso.inv_comp_assoc {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {h : Y ⟶ Z} [IsIso f] [IsIso h] {Z✝ : C} (h✝ : X ⟶ Z✝) : CategoryStruct.comp (inv (CategoryStruct.comp f h)) h✝ = CategoryStruct.comp (inv h) (CategoryStruct.comp (inv f) h✝)

@[simp]

theorem CategoryTheory.IsIso.inv_inv {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ⟶ Y} [IsIso f] : inv (inv f) = f

@[simp]

theorem CategoryTheory.IsIso.Iso.inv_inv {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ≅ Y) : inv f.inv = f.hom

@[simp]

theorem CategoryTheory.IsIso.Iso.inv_hom {C : Type u} [Category.{v, u} C] {X Y : C} (f : X ≅ Y) : inv f.hom = f.inv

@[simp]

theorem CategoryTheory.IsIso.inv_comp_eq {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ⟶ Y) [IsIso α] {f : X ⟶ Z} {g : Y ⟶ Z} : CategoryStruct.comp (inv α) f = g ↔ f = CategoryStruct.comp α g

@[simp]

theorem CategoryTheory.IsIso.comp_inv_eq {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : Y ⟶ X) [IsIso α] {f : Z ⟶ X} {g : Z ⟶ Y} : CategoryStruct.comp f (inv α) = g ↔ f = CategoryStruct.comp g α

@[simp]

theorem CategoryTheory.IsIso.eq_inv_comp {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : X ⟶ Y) [IsIso α] {f : X ⟶ Z} {g : Y ⟶ Z} : g = CategoryStruct.comp (inv α) f ↔ CategoryStruct.comp α g = f

@[simp]

theorem CategoryTheory.IsIso.eq_comp_inv {C : Type u} [Category.{v, u} C] {X Y Z : C} (α : Y ⟶ X) [IsIso α] {f : Z ⟶ X} {g : Z ⟶ Y} : g = CategoryStruct.comp f (inv α) ↔ CategoryStruct.comp g α = f

theorem CategoryTheory.IsIso.of_isIso_comp_left {C : Type u} [Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] [IsIso (CategoryStruct.comp f g)] : IsIso g

theorem CategoryTheory.IsIso.of_isIso_comp_right {C : Type u} [Category.{v, u} C] {X Y Z : C} (g : Z ⟶ Y) (f : Y ⟶ X) [IsIso f] [IsIso (CategoryStruct.comp g f)] : IsIso g

theorem CategoryTheory.IsIso.of_isIso_fac_left {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [IsIso f] [hh : IsIso h] (w : CategoryStruct.comp f g = h) : IsIso g

theorem CategoryTheory.IsIso.of_isIso_fac_right {C : Type u} [Category.{v, u} C] {X Y Z : C} {f : Y ⟶ X} {g : Z ⟶ Y} {h : Z ⟶ X} [IsIso f] [hh : IsIso h] (w : CategoryStruct.comp g f = h) : IsIso g

@[simp]

theorem CategoryTheory.isIso_comp_left_iff {C : Type u} [Category.{v, u} C] {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : IsIso (CategoryStruct.comp f g) ↔ IsIso g

@[simp]

theorem CategoryTheory.isIso_comp_right_iff {C : Type u} [Category.{v, u} C] {X Y Z : C} (g : Z ⟶ Y) (f : Y ⟶ X) [IsIso f] : IsIso (CategoryStruct.comp g f) ↔ IsIso g

theorem CategoryTheory.eq_of_inv_eq_inv {C : Type u} [Category.{v, u} C] {X Y : C} {f g : X ⟶ Y} [IsIso f] [IsIso g] (p : inv f = inv g) : f = g

theorem CategoryTheory.IsIso.inv_eq_inv {C : Type u} [Category.{v, u} C] {X Y : C} {f g : X ⟶ Y} [IsIso f] [IsIso g] : inv f = inv g ↔ f = g

theorem CategoryTheory.hom_comp_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : X ⟶ Y) [IsIso g] {f : Y ⟶ X} : CategoryStruct.comp g f = CategoryStruct.id X ↔ f = inv g

theorem CategoryTheory.comp_hom_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : Y ⟶ X) [IsIso g] {f : X ⟶ Y} : CategoryStruct.comp f g = CategoryStruct.id X ↔ f = inv g

theorem CategoryTheory.inv_comp_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : X ⟶ Y) [IsIso g] {f : X ⟶ Y} : CategoryStruct.comp (inv g) f = CategoryStruct.id Y ↔ f = g

theorem CategoryTheory.comp_inv_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : Y ⟶ X) [IsIso g] {f : Y ⟶ X} : CategoryStruct.comp f (inv g) = CategoryStruct.id Y ↔ f = g

theorem CategoryTheory.isIso_of_hom_comp_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : X ⟶ Y) [IsIso g] {f : Y ⟶ X} (h : CategoryStruct.comp g f = CategoryStruct.id X) : IsIso f

theorem CategoryTheory.isIso_of_comp_hom_eq_id {C : Type u} [Category.{v, u} C] {X Y : C} (g : Y ⟶ X) [IsIso g] {f : X ⟶ Y} (h : CategoryStruct.comp f g = CategoryStruct.id X) : IsIso f

theorem CategoryTheory.Iso.inv_ext {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ≅ Y} {g : Y ⟶ X} (hom_inv_id : CategoryStruct.comp f.hom g = CategoryStruct.id X) : f.inv = g

theorem CategoryTheory.Iso.inv_ext' {C : Type u} [Category.{v, u} C] {X Y : C} {f : X ≅ Y} {g : Y ⟶ X} (hom_inv_id : CategoryStruct.comp f.hom g = CategoryStruct.id X) : g = f.inv

All these cancellation lemmas can be solved by simp [cancel_mono] (or simp [cancel_epi]), but with the current design cancel_mono is not a good simp lemma, because it generates a typeclass search.

When we can see syntactically that a morphism is a mono or an epi because it came from an isomorphism, it's fine to do the cancellation via simp.

In the longer term, it might be worth exploring making mono and epi structures, rather than typeclasses, with coercions back to X ⟶ Y. Presumably we could write X ↪ Y and X ↠ Y.

@[simp]

theorem CategoryTheory.Iso.cancel_iso_hom_left {C : Type u} [Category.{v, u} C] {X Y Z : C} (f : X ≅ Y) (g g' : Y ⟶ Z) : CategoryStruct.comp f.hom g = CategoryStruct.comp f.hom g' ↔ g = g'

@[simp]

theorem CategoryTheory.Iso.cancel_iso_inv_right {C : Type u} [Category.{v, u} C] {X Y Z : C} (g g' : Z ⟶ Y) (f : X ≅ Y) : CategoryStruct.comp g f.inv = CategoryStruct.comp g' f.inv ↔ g = g'

@[simp]

theorem CategoryTheory.Iso.cancel_iso_hom_right {C : Type u} [Category.{v, u} C] {X Y Z : C} (f f' : X ⟶ Y) (g : Y ≅ Z) : CategoryStruct.comp f g.hom = CategoryStruct.comp f' g.hom ↔ f = f'

@[simp]

theorem CategoryTheory.Iso.cancel_iso_inv_left {C : Type u} [Category.{v, u} C] {X Y Z : C} (g : Y ≅ Z) (f f' : Y ⟶ X) : CategoryStruct.comp g.inv f = CategoryStruct.comp g.inv f' ↔ f = f'

@[simp]

theorem CategoryTheory.Iso.cancel_iso_hom_right_assoc {C : Type u} [Category.{v, u} C] {W X X' Y Z : C} (f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y) (h : Y ≅ Z) : CategoryStruct.comp f (CategoryStruct.comp g h.hom) = CategoryStruct.comp f' (CategoryStruct.comp g' h.hom) ↔ CategoryStruct.comp f g = CategoryStruct.comp f' g'

@[simp]

theorem CategoryTheory.Iso.cancel_iso_inv_right_assoc {C : Type u} [Category.{v, u} C] {W X X' Y Z : C} (f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y) (h : Z ≅ Y) : CategoryStruct.comp f (CategoryStruct.comp g h.inv) = CategoryStruct.comp f' (CategoryStruct.comp g' h.inv) ↔ CategoryStruct.comp f g = CategoryStruct.comp f' g'

@[simp]

theorem CategoryTheory.Iso.map_hom_inv_id {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{v_1, u_1} D] {X Y : C} (e : X ≅ Y) (F : Functor C D) : CategoryStruct.comp (F.map e.hom) (F.map e.inv) = CategoryStruct.id (F.obj X)

@[simp]

theorem CategoryTheory.Iso.map_hom_inv_id_assoc {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{v_1, u_1} D] {X Y : C} (e : X ≅ Y) (F : Functor C D) {Z : D} (h : F.obj X ⟶ Z) : CategoryStruct.comp (F.map e.hom) (CategoryStruct.comp (F.map e.inv) h) = h

@[simp]

theorem CategoryTheory.Iso.map_inv_hom_id {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{v_1, u_1} D] {X Y : C} (e : X ≅ Y) (F : Functor C D) : CategoryStruct.comp (F.map e.inv) (F.map e.hom) = CategoryStruct.id (F.obj Y)

@[simp]

theorem CategoryTheory.Iso.map_inv_hom_id_assoc {C : Type u} [Category.{v, u} C] {D : Type u_1} [Category.{v_1, u_1} D] {X Y : C} (e : X ≅ Y) (F : Functor C D) {Z : D} (h : F.obj Y ⟶ Z) : CategoryStruct.comp (F.map e.inv) (CategoryStruct.comp (F.map e.hom) h) = h

def CategoryTheory.Functor.mapIso {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (i : X ≅ Y) : F.obj X ≅ F.obj Y

A functor F : C ⥤ D sends isomorphisms i : X ≅ Y to isomorphisms F.obj X ≅ F.obj Y

@[simp]

theorem CategoryTheory.Functor.mapIso_hom {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (i : X ≅ Y) : (F.mapIso i).hom = F.map i.hom

@[simp]

theorem CategoryTheory.Functor.mapIso_inv {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (i : X ≅ Y) : (F.mapIso i).inv = F.map i.inv

@[simp]

theorem CategoryTheory.Functor.mapIso_symm {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (i : X ≅ Y) : F.mapIso i.symm = (F.mapIso i).symm

@[simp]

theorem CategoryTheory.Functor.mapIso_trans {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y Z : C} (i : X ≅ Y) (j : Y ≅ Z) : F.mapIso (i ≪≫ j) = F.mapIso i ≪≫ F.mapIso j

@[simp]

theorem CategoryTheory.Functor.mapIso_refl {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (X : C) : F.mapIso (Iso.refl X) = Iso.refl (F.obj X)

instance CategoryTheory.Functor.map_isIso {C : Type u} [Category.{v, u} C] {X Y : C} {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) (f : X ⟶ Y) [IsIso f] : IsIso (F.map f)

@[simp]

theorem CategoryTheory.Functor.map_inv {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (f : X ⟶ Y) [IsIso f] : F.map (CategoryTheory.inv f) = CategoryTheory.inv (F.map f)

theorem CategoryTheory.Functor.map_hom_inv {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (f : X ⟶ Y) [IsIso f] : CategoryStruct.comp (F.map f) (F.map (CategoryTheory.inv f)) = CategoryStruct.id (F.obj X)

theorem CategoryTheory.Functor.map_inv_hom {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (f : Y ⟶ X) [IsIso f] : CategoryStruct.comp (F.map (CategoryTheory.inv f)) (F.map f) = CategoryStruct.id (F.obj X)

theorem CategoryTheory.Functor.map_hom_inv_assoc {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (f : X ⟶ Y) [IsIso f] {Z : D} (h : F.obj X ⟶ Z) : CategoryStruct.comp (F.map f) (CategoryStruct.comp (F.map (CategoryTheory.inv f)) h) = h

theorem CategoryTheory.Functor.map_inv_hom_assoc {C : Type u} [Category.{v, u} C] {D : Type u₂} [Category.{v₂, u₂} D] (F : Functor C D) {X Y : C} (f : Y ⟶ X) [IsIso f] {Z : D} (h : F.obj X ⟶ Z) : CategoryStruct.comp (F.map (CategoryTheory.inv f)) (CategoryStruct.comp (F.map f) h) = h