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groupoid.inv_eq_inv (f : X ⟶ Y) : groupoid.inv f = inv f | is_iso.eq_inv_of_hom_inv_id $ groupoid.comp_inv f | lemma | category_theory.groupoid.inv_eq_inv | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid.inv_equiv : (X ⟶ Y) ≃ (Y ⟶ X) | ⟨groupoid.inv, groupoid.inv, λ f, by simp, λ f, by simp⟩ | def | category_theory.groupoid.inv_equiv | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | `groupoid.inv` is involutive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid_has_involutive_reverse : quiver.has_involutive_reverse C | { reverse' := λ X Y f, groupoid.inv f,
inv' := λ X Y f, by { dsimp [quiver.reverse], simp, } } | instance | category_theory.groupoid_has_involutive_reverse | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"quiver.has_involutive_reverse",
"quiver.reverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid.reverse_eq_inv (f : X ⟶ Y) : quiver.reverse f = groupoid.inv f | rfl | lemma | category_theory.groupoid.reverse_eq_inv | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"quiver.reverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
functor_map_reverse {D : Type*} [groupoid D] (F : C ⥤ D) :
F.to_prefunctor.map_reverse | { map_reverse' := λ X Y f, by
simp only [quiver.reverse, quiver.has_reverse.reverse', groupoid.inv_eq_inv,
functor.to_prefunctor_map, functor.map_inv], } | instance | category_theory.functor_map_reverse | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"quiver.reverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid.iso_equiv_hom : (X ≅ Y) ≃ (X ⟶ Y) | { to_fun := iso.hom,
inv_fun := λ f, ⟨f, groupoid.inv f⟩,
left_inv := λ i, iso.ext rfl,
right_inv := λ f, rfl } | def | category_theory.groupoid.iso_equiv_hom | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"inv_fun"
] | In a groupoid, isomorphisms are equivalent to morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid.inv_functor : C ⥤ Cᵒᵖ | { obj := opposite.op,
map := λ {X Y} f, (inv f).op } | def | category_theory.groupoid.inv_functor | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"opposite.op"
] | The functor from a groupoid `C` to its opposite sending every morphism to its inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid.of_is_iso (all_is_iso : ∀ {X Y : C} (f : X ⟶ Y), is_iso f) : groupoid.{v} C | { inv := λ X Y f, inv f } | def | category_theory.groupoid.of_is_iso | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | A category where every morphism `is_iso` is a groupoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
groupoid.of_hom_unique (all_unique : ∀ {X Y : C}, unique (X ⟶ Y)) : groupoid.{v} C | { inv := λ X Y f, all_unique.default } | def | category_theory.groupoid.of_hom_unique | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [
"unique"
] | A category with a unique morphism between any two objects is a groupoid | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_category.groupoid {C : Type u} (D : Type u₂) [groupoid.{v} D] (F : C → D) :
groupoid.{v} (induced_category D F) | { inv := λ X Y f, groupoid.inv f,
inv_comp' := λ X Y f, groupoid.inv_comp f,
comp_inv' := λ X Y f, groupoid.comp_inv f,
.. induced_category.category F } | instance | category_theory.induced_category.groupoid | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid_pi {I : Type u} {J : I → Type u₂} [∀ i, groupoid.{v} (J i)] :
groupoid.{max u v} (Π i : I, J i) | { inv := λ (x y : Π i, J i) (f : Π i, x i ⟶ y i), (λ i : I, groupoid.inv (f i)), } | instance | category_theory.groupoid_pi | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
groupoid_prod {α : Type u} {β : Type v} [groupoid.{u₂} α] [groupoid.{v₂} β] :
groupoid.{max u₂ v₂} (α × β) | { inv := λ (x y : α × β) (f : x ⟶ y), (groupoid.inv f.1, groupoid.inv f.2) } | instance | category_theory.groupoid_prod | category_theory | src/category_theory/groupoid.lean | [
"category_theory.full_subcategory",
"category_theory.products.basic",
"category_theory.pi.basic",
"category_theory.category.basic",
"combinatorics.quiver.connected_component"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso {C : Type u} [category.{v} C] (X Y : C) | (hom : X ⟶ Y)
(inv : Y ⟶ X)
(hom_inv_id' : hom ≫ inv = 𝟙 X . obviously)
(inv_hom_id' : inv ≫ hom = 𝟙 Y . obviously) | structure | category_theory.iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [
"iso"
] | An isomorphism (a.k.a. an invertible morphism) between two objects of a category.
The inverse morphism is bundled.
See also `category_theory.core` for the category with the same objects and isomorphisms playing
the role of morphisms.
See <https://stacks.math.columbia.edu/tag/0017>. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext ⦃α β : X ≅ Y⦄ (w : α.hom = β.hom) : α = β | suffices α.inv = β.inv, by cases α; cases β; cc,
calc α.inv
= α.inv ≫ (β.hom ≫ β.inv) : by rw [iso.hom_inv_id, category.comp_id]
... = (α.inv ≫ α.hom) ≫ β.inv : by rw [category.assoc, ←w]
... = β.inv : by rw [iso.inv_hom_id, category.id_comp] | lemma | category_theory.iso.ext | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (I : X ≅ Y) : Y ≅ X | { hom := I.inv,
inv := I.hom,
hom_inv_id' := I.inv_hom_id',
inv_hom_id' := I.hom_inv_id' } | def | category_theory.iso.symm | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | Inverse isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_hom (α : X ≅ Y) : α.symm.hom = α.inv | rfl | lemma | category_theory.iso.symm_hom | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_inv (α : X ≅ Y) : α.symm.inv = α.hom | rfl | lemma | category_theory.iso.symm_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk {X Y : C} (hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id) (inv_hom_id) :
iso.symm {hom := hom, inv := inv, hom_inv_id' := hom_inv_id, inv_hom_id' := inv_hom_id} =
{hom := inv, inv := hom, hom_inv_id' := inv_hom_id, inv_hom_id' := hom_inv_id} | rfl | lemma | category_theory.iso.symm_mk | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm_eq {X Y : C} (α : X ≅ Y) : α.symm.symm = α | by cases α; refl | lemma | category_theory.iso.symm_symm_eq | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_eq_iff {X Y : C} {α β : X ≅ Y} : α.symm = β.symm ↔ α = β | ⟨λ h, symm_symm_eq α ▸ symm_symm_eq β ▸ congr_arg symm h, congr_arg symm⟩ | lemma | category_theory.iso.symm_eq_iff | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_iso_symm (X Y : C) : nonempty (X ≅ Y) ↔ nonempty (Y ≅ X) | ⟨λ h, ⟨h.some.symm⟩, λ h, ⟨h.some.symm⟩⟩ | lemma | category_theory.iso.nonempty_iso_symm | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (X : C) : X ≅ X | { hom := 𝟙 X,
inv := 𝟙 X } | def | category_theory.iso.refl | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | Identity isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_symm (X : C) : (iso.refl X).symm = iso.refl X | rfl | lemma | category_theory.iso.refl_symm | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (α : X ≅ Y) (β : Y ≅ Z) : X ≅ Z | { hom := α.hom ≫ β.hom,
inv := β.inv ≫ α.inv } | def | category_theory.iso.trans | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | Composition of two isomorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_mk {X Y Z : C}
(hom : X ⟶ Y) (inv : Y ⟶ X) (hom_inv_id) (inv_hom_id)
(hom' : Y ⟶ Z) (inv' : Z ⟶ Y) (hom_inv_id') (inv_hom_id') (hom_inv_id'') (inv_hom_id'') :
iso.trans
{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... | rfl | lemma | category_theory.iso.trans_mk | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_symm (α : X ≅ Y) (β : Y ≅ Z) : (α ≪≫ β).symm = β.symm ≪≫ α.symm | rfl | lemma | category_theory.iso.trans_symm | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_assoc {Z' : C} (α : X ≅ Y) (β : Y ≅ Z) (γ : Z ≅ Z') :
(α ≪≫ β) ≪≫ γ = α ≪≫ β ≪≫ γ | by ext; simp only [trans_hom, category.assoc] | lemma | category_theory.iso.trans_assoc | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_trans (α : X ≅ Y) : (iso.refl X) ≪≫ α = α | by ext; apply category.id_comp | lemma | category_theory.iso.refl_trans | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_refl (α : X ≅ Y) : α ≪≫ (iso.refl Y) = α | by ext; apply category.comp_id | lemma | category_theory.iso.trans_refl | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_self_id (α : X ≅ Y) : α.symm ≪≫ α = iso.refl Y | ext α.inv_hom_id | lemma | category_theory.iso.symm_self_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_symm_id (α : X ≅ Y) : α ≪≫ α.symm = iso.refl X | ext α.hom_inv_id | lemma | category_theory.iso.self_symm_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_self_id_assoc (α : X ≅ Y) (β : Y ≅ Z) : α.symm ≪≫ α ≪≫ β = β | by rw [← trans_assoc, symm_self_id, refl_trans] | lemma | category_theory.iso.symm_self_id_assoc | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_symm_id_assoc (α : X ≅ Y) (β : X ≅ Z) : α ≪≫ α.symm ≪≫ β = β | by rw [← trans_assoc, self_symm_id, refl_trans] | lemma | category_theory.iso.self_symm_id_assoc | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_eq (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : α.inv ≫ f = g ↔ f = α.hom ≫ g | ⟨λ H, by simp [H.symm], λ H, by simp [H]⟩ | lemma | category_theory.iso.inv_comp_eq | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_comp (α : X ≅ Y) {f : X ⟶ Z} {g : Y ⟶ Z} : g = α.inv ≫ f ↔ α.hom ≫ g = f | (inv_comp_eq α.symm).symm | lemma | category_theory.iso.eq_inv_comp | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv_eq (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : f ≫ α.inv = g ↔ f = g ≫ α.hom | ⟨λ H, by simp [H.symm], λ H, by simp [H]⟩ | lemma | category_theory.iso.comp_inv_eq | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_inv (α : X ≅ Y) {f : Z ⟶ Y} {g : Z ⟶ X} : g = f ≫ α.inv ↔ g ≫ α.hom = f | (comp_inv_eq α.symm).symm | lemma | category_theory.iso.eq_comp_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_inv (f g : X ≅ Y) : f.inv = g.inv ↔ f.hom = g.hom | have ∀{X Y : C} (f g : X ≅ Y), f.hom = g.hom → f.inv = g.inv, from λ X Y f g h, by rw [ext h],
⟨this f.symm g.symm, this f g⟩ | lemma | category_theory.iso.inv_eq_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_comp_eq_id (α : X ≅ Y) {f : Y ⟶ X} : α.hom ≫ f = 𝟙 X ↔ f = α.inv | by rw [←eq_inv_comp, comp_id] | lemma | category_theory.iso.hom_comp_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom_eq_id (α : X ≅ Y) {f : Y ⟶ X} : f ≫ α.hom = 𝟙 Y ↔ f = α.inv | by rw [←eq_comp_inv, id_comp] | lemma | category_theory.iso.comp_hom_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_eq_id (α : X ≅ Y) {f : X ⟶ Y} : α.inv ≫ f = 𝟙 Y ↔ f = α.hom | hom_comp_eq_id α.symm | lemma | category_theory.iso.inv_comp_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv_eq_id (α : X ≅ Y) {f : X ⟶ Y} : f ≫ α.inv = 𝟙 X ↔ f = α.hom | comp_hom_eq_id α.symm | lemma | category_theory.iso.comp_inv_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_eq_inv (α : X ≅ Y) (β : Y ≅ X) : α.hom = β.inv ↔ β.hom = α.inv | by { erw [inv_eq_inv α.symm β, eq_comm], refl } | lemma | category_theory.iso.hom_eq_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso (f : X ⟶ Y) : Prop | (out : ∃ inv : Y ⟶ X, f ≫ inv = 𝟙 X ∧ inv ≫ f = 𝟙 Y) | class | category_theory.is_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | `is_iso` typeclass expressing that a morphism is invertible. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv (f : X ⟶ Y) [I : is_iso f] | classical.some I.1 | def | category_theory.inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | The inverse of a morphism `f` when we have `[is_iso f]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_inv_id (f : X ⟶ Y) [I : is_iso f] : f ≫ inv f = 𝟙 X | (classical.some_spec I.1).left | lemma | category_theory.is_iso.hom_inv_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_hom_id (f : X ⟶ Y) [I : is_iso f] : inv f ≫ f = 𝟙 Y | (classical.some_spec I.1).right | lemma | category_theory.is_iso.inv_hom_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_iso (f : X ⟶ Y) [h : is_iso f] : X ≅ Y | ⟨f, inv f, hom_inv_id f, inv_hom_id f⟩ | def | category_theory.as_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | Reinterpret a morphism `f` with an `is_iso f` instance as an `iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
as_iso_hom (f : X ⟶ Y) [is_iso f] : (as_iso f).hom = f | rfl | lemma | category_theory.as_iso_hom | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
as_iso_inv (f : X ⟶ Y) [is_iso f] : (as_iso f).inv = inv f | rfl | lemma | category_theory.as_iso_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
epi_of_iso (f : X ⟶ Y) [is_iso f] : epi f | { left_cancellation := λ Z g h w,
-- This is an interesting test case for better rewrite automation.
by rw [← is_iso.inv_hom_id_assoc f g, w, is_iso.inv_hom_id_assoc f h] } | instance | category_theory.is_iso.epi_of_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_of_iso (f : X ⟶ Y) [is_iso f] : mono f | { right_cancellation := λ Z g h w,
by rw [← category.comp_id g, ← category.comp_id h, ← is_iso.hom_inv_id f, ← category.assoc, w,
← category.assoc] } | instance | category_theory.is_iso.mono_of_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_hom_inv_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(hom_inv_id : f ≫ g = 𝟙 X) : inv f = g | begin
apply (cancel_epi f).mp,
simp [hom_inv_id],
end | lemma | category_theory.is_iso.inv_eq_of_hom_inv_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_of_inv_hom_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(inv_hom_id : g ≫ f = 𝟙 Y) : inv f = g | begin
apply (cancel_mono f).mp,
simp [inv_hom_id],
end | lemma | category_theory.is_iso.inv_eq_of_inv_hom_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_hom_inv_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(hom_inv_id : f ≫ g = 𝟙 X) : g = inv f | (inv_eq_of_hom_inv_id hom_inv_id).symm | lemma | category_theory.is_iso.eq_inv_of_hom_inv_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_of_inv_hom_id {f : X ⟶ Y} [is_iso f] {g : Y ⟶ X}
(inv_hom_id : g ≫ f = 𝟙 Y) : g = inv f | (inv_eq_of_inv_hom_id inv_hom_id).symm | lemma | category_theory.is_iso.eq_inv_of_inv_hom_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id (X : C) : is_iso (𝟙 X) | ⟨⟨𝟙 X, by simp⟩⟩ | instance | category_theory.is_iso.id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso (f : X ≅ Y) : is_iso f.hom | ⟨⟨f.inv, by simp⟩⟩ | instance | category_theory.is_iso.of_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_iso_inv (f : X ≅ Y) : is_iso f.inv | is_iso.of_iso f.symm | instance | category_theory.is_iso.of_iso_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_is_iso [is_iso f] : is_iso (inv f) | is_iso.of_iso_inv (as_iso f) | instance | category_theory.is_iso.inv_is_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_is_iso [is_iso f] [is_iso h] : is_iso (f ≫ h) | is_iso.of_iso $ (as_iso f) ≪≫ (as_iso h) | instance | category_theory.is_iso.comp_is_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_id : inv (𝟙 X) = 𝟙 X | by { ext, simp, } | lemma | category_theory.is_iso.inv_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp [is_iso f] [is_iso h] : inv (f ≫ h) = inv h ≫ inv f | by { ext, simp, } | lemma | category_theory.is_iso.inv_comp | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv [is_iso f] : inv (inv f) = f | by { ext, simp, } | lemma | category_theory.is_iso.inv_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.inv_inv (f : X ≅ Y) : inv (f.inv) = f.hom | by { ext, simp, } | lemma | category_theory.is_iso.iso.inv_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iso.inv_hom (f : X ≅ Y) : inv (f.hom) = f.inv | by { ext, simp, } | lemma | category_theory.is_iso.iso.inv_hom | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_eq (α : X ⟶ Y) [is_iso α] {f : X ⟶ Z} {g : Y ⟶ Z} : inv α ≫ f = g ↔ f = α ≫ g | (as_iso α).inv_comp_eq | lemma | category_theory.is_iso.inv_comp_eq | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_comp (α : X ⟶ Y) [is_iso α] {f : X ⟶ Z} {g : Y ⟶ Z} : g = inv α ≫ f ↔ α ≫ g = f | (as_iso α).eq_inv_comp | lemma | category_theory.is_iso.eq_inv_comp | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv_eq (α : X ⟶ Y) [is_iso α] {f : Z ⟶ Y} {g : Z ⟶ X} : f ≫ inv α = g ↔ f = g ≫ α | (as_iso α).comp_inv_eq | lemma | category_theory.is_iso.comp_inv_eq | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_inv (α : X ⟶ Y) [is_iso α] {f : Z ⟶ Y} {g : Z ⟶ X} : g = f ≫ inv α ↔ g ≫ α = f | (as_iso α).eq_comp_inv | lemma | category_theory.is_iso.eq_comp_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso_comp_left {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[is_iso f] [is_iso (f ≫ g)] : is_iso g | by { rw [← id_comp g, ← inv_hom_id f, assoc], apply_instance, } | lemma | category_theory.is_iso.of_is_iso_comp_left | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso_comp_right {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z)
[is_iso g] [is_iso (f ≫ g)] : is_iso f | by { rw [← comp_id f, ← hom_inv_id g, ← assoc], apply_instance, } | lemma | category_theory.is_iso.of_is_iso_comp_right | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso_fac_left {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z}
[is_iso f] [hh : is_iso h] (w : f ≫ g = h) : is_iso g | by { rw ← w at hh, haveI := hh, exact of_is_iso_comp_left f g, } | lemma | category_theory.is_iso.of_is_iso_fac_left | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_is_iso_fac_right {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z}
[is_iso g] [hh : is_iso h] (w : f ≫ g = h) : is_iso f | by { rw ← w at hh, haveI := hh, exact of_is_iso_comp_right f g, } | lemma | category_theory.is_iso.of_is_iso_fac_right | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_inv_eq_inv {f g : X ⟶ Y} [is_iso f] [is_iso g] (p : inv f = inv g) : f = g | begin
apply (cancel_epi (inv f)).1,
erw [inv_hom_id, p, inv_hom_id],
end | lemma | category_theory.eq_of_inv_eq_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso.inv_eq_inv {f g : X ⟶ Y} [is_iso f] [is_iso g] : inv f = inv g ↔ f = g | iso.inv_eq_inv (as_iso f) (as_iso g) | lemma | category_theory.is_iso.inv_eq_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hom_comp_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} : g ≫ f = 𝟙 X ↔ f = inv g | (as_iso g).hom_comp_eq_id | lemma | category_theory.hom_comp_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_hom_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} : f ≫ g = 𝟙 Y ↔ f = inv g | (as_iso g).comp_hom_eq_id | lemma | category_theory.comp_hom_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp_eq_id (g : X ⟶ Y) [is_iso g] {f : X ⟶ Y} : inv g ≫ f = 𝟙 Y ↔ f = g | (as_iso g).inv_comp_eq_id | lemma | category_theory.inv_comp_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv_eq_id (g : X ⟶ Y) [is_iso g] {f : X ⟶ Y} : f ≫ inv g = 𝟙 X ↔ f = g | (as_iso g).comp_inv_eq_id | lemma | category_theory.comp_inv_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_hom_comp_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} (h : g ≫ f = 𝟙 X) : is_iso f | by { rw [(hom_comp_eq_id _).mp h], apply_instance } | lemma | category_theory.is_iso_of_hom_comp_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_iso_of_comp_hom_eq_id (g : X ⟶ Y) [is_iso g] {f : Y ⟶ X} (h : f ≫ g = 𝟙 Y) : is_iso f | by { rw [(comp_hom_eq_id _).mp h], apply_instance } | lemma | category_theory.is_iso_of_comp_hom_eq_id | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_ext {f : X ≅ Y} {g : Y ⟶ X}
(hom_inv_id : f.hom ≫ g = 𝟙 X) : f.inv = g | ((hom_comp_eq_id f).1 hom_inv_id).symm | lemma | category_theory.iso.inv_ext | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_ext' {f : X ≅ Y} {g : Y ⟶ X}
(hom_inv_id : f.hom ≫ g = 𝟙 X) : g = f.inv | (hom_comp_eq_id f).1 hom_inv_id | lemma | category_theory.iso.inv_ext' | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_hom_left {X Y Z : C} (f : X ≅ Y) (g g' : Y ⟶ Z) :
f.hom ≫ g = f.hom ≫ g' ↔ g = g' | by simp only [cancel_epi] | lemma | category_theory.iso.cancel_iso_hom_left | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_inv_left {X Y Z : C} (f : Y ≅ X) (g g' : Y ⟶ Z) :
f.inv ≫ g = f.inv ≫ g' ↔ g = g' | by simp only [cancel_epi] | lemma | category_theory.iso.cancel_iso_inv_left | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_hom_right {X Y Z : C} (f f' : X ⟶ Y) (g : Y ≅ Z) :
f ≫ g.hom = f' ≫ g.hom ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.iso.cancel_iso_hom_right | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_inv_right {X Y Z : C} (f f' : X ⟶ Y) (g : Z ≅ Y) :
f ≫ g.inv = f' ≫ g.inv ↔ f = f' | by simp only [cancel_mono] | lemma | category_theory.iso.cancel_iso_inv_right | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_hom_right_assoc {W X X' Y Z : C}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y)
(h : Y ≅ Z) :
f ≫ g ≫ h.hom = f' ≫ g' ≫ h.hom ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.iso.cancel_iso_hom_right_assoc | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_iso_inv_right_assoc {W X X' Y Z : C}
(f : W ⟶ X) (g : X ⟶ Y) (f' : W ⟶ X') (g' : X' ⟶ Y)
(h : Z ≅ Y) :
f ≫ g ≫ h.inv = f' ≫ g' ≫ h.inv ↔ f ≫ g = f' ≫ g' | by simp only [←category.assoc, cancel_mono] | lemma | category_theory.iso.cancel_iso_inv_right_assoc | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso (F : C ⥤ D) {X Y : C} (i : X ≅ Y) : F.obj X ≅ F.obj Y | { hom := F.map i.hom,
inv := F.map i.inv,
hom_inv_id' := by rw [←map_comp, iso.hom_inv_id, ←map_id],
inv_hom_id' := by rw [←map_comp, iso.inv_hom_id, ←map_id] } | def | category_theory.functor.map_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | A functor `F : C ⥤ D` sends isomorphisms `i : X ≅ Y` to isomorphisms `F.obj X ≅ F.obj Y` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_iso_symm (F : C ⥤ D) {X Y : C} (i : X ≅ Y) :
F.map_iso i.symm = (F.map_iso i).symm | rfl | lemma | category_theory.functor.map_iso_symm | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_trans (F : C ⥤ D) {X Y Z : C} (i : X ≅ Y) (j : Y ≅ Z) :
F.map_iso (i ≪≫ j) = (F.map_iso i) ≪≫ (F.map_iso j) | by ext; apply functor.map_comp | lemma | category_theory.functor.map_iso_trans | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_iso_refl (F : C ⥤ D) (X : C) : F.map_iso (iso.refl X) = iso.refl (F.obj X) | iso.ext $ F.map_id X | lemma | category_theory.functor.map_iso_refl | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_is_iso (F : C ⥤ D) (f : X ⟶ Y) [is_iso f] : is_iso (F.map f) | is_iso.of_iso $ F.map_iso (as_iso f) | instance | category_theory.functor.map_is_iso | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map (inv f) = inv (F.map f) | by { ext, simp [←F.map_comp], } | lemma | category_theory.functor.map_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [
"map_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) | by simp | lemma | category_theory.functor.map_hom_inv | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv_hom (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [is_iso f] :
F.map (inv f) ≫ F.map f = 𝟙 (F.obj Y) | by simp | lemma | category_theory.functor.map_inv_hom | category_theory | src/category_theory/isomorphism.lean | [
"category_theory.functor.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_isomorphic : C → C → Prop | λ X Y, nonempty (X ≅ Y) | def | category_theory.is_isomorphic | category_theory | src/category_theory/isomorphism_classes.lean | [
"category_theory.category.Cat",
"category_theory.groupoid",
"category_theory.types"
] | [] | An object `X` is isomorphic to an object `Y`, if `X ≅ Y` is not empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_isomorphic_setoid : setoid C | { r := is_isomorphic,
iseqv := ⟨λ X, ⟨iso.refl X⟩, λ X Y ⟨α⟩, ⟨α.symm⟩, λ X Y Z ⟨α⟩ ⟨β⟩, ⟨α.trans β⟩⟩ } | def | category_theory.is_isomorphic_setoid | category_theory | src/category_theory/isomorphism_classes.lean | [
"category_theory.category.Cat",
"category_theory.groupoid",
"category_theory.types"
] | [] | `is_isomorphic` defines a setoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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