Documentation

Mathlib.CategoryTheory.Groupoid

Groupoids #

We define Groupoid as a typeclass extending Category, asserting that all morphisms have inverses.

The instance IsIso.ofGroupoid (f : X ⟶ Y) : IsIso f means that you can then write inv f to access the inverse of any morphism f.

Groupoid.isoEquivHom : (X ≅ Y) ≃ (X ⟶ Y) provides the equivalence between isomorphisms and morphisms in a groupoid.

We provide a (non-instance) constructor Groupoid.ofIsIso from an existing category with IsIso f for every f.

See also #

See also CategoryTheory.Core for the groupoid of isomorphisms in a category.

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class CategoryTheory.Groupoid (obj : Type u) extends CategoryTheory.Category :
Type (max u (v + 1))

A Groupoid is a category such that all morphisms are isomorphisms.

Instances
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theorem CategoryTheory.Groupoid.inv_comp {obj : Type u} [self : CategoryTheory.Groupoid obj] {X : obj} {Y : obj} (f : X ⟶ Y) :
CategoryTheory.CategoryStruct.comp (CategoryTheory.Groupoid.inv f) f = CategoryTheory.CategoryStruct.id Y

inv f composed f is the identity

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theorem CategoryTheory.Groupoid.comp_inv {obj : Type u} [self : CategoryTheory.Groupoid obj] {X : obj} {Y : obj} (f : X ⟶ Y) :
CategoryTheory.CategoryStruct.comp f (CategoryTheory.Groupoid.inv f) = CategoryTheory.CategoryStruct.id X

f composed with inv f is the identity

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@[reducible, inline]
abbrev CategoryTheory.LargeGroupoid (C : Type (u + 1)) :
Type (u + 1)

A LargeGroupoid is a groupoid where the objects live in Type (u+1) while the morphisms live in Type u.

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@[reducible, inline]
abbrev CategoryTheory.SmallGroupoid (C : Type u) :
Type (u + 1)

A SmallGroupoid is a groupoid where the objects and morphisms live in the same universe.

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@[instance 100]
instance CategoryTheory.IsIso.of_groupoid {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} (f : X ⟶ Y) :
CategoryTheory.IsIso f
Equations
  • ⋯ = ⋯
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@[simp]
theorem CategoryTheory.Groupoid.inv_eq_inv {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} (f : X ⟶ Y) :
CategoryTheory.Groupoid.inv f = CategoryTheory.inv f
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@[simp]
theorem CategoryTheory.Groupoid.invEquiv_apply {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} :
∀ (a : X ⟶ Y), CategoryTheory.Groupoid.invEquiv a = CategoryTheory.Groupoid.inv a
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@[simp]
theorem CategoryTheory.Groupoid.invEquiv_symm_apply {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} :
∀ (a : Y ⟶ X), CategoryTheory.Groupoid.invEquiv.symm a = CategoryTheory.Groupoid.inv a
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def CategoryTheory.Groupoid.invEquiv {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} :
(X ⟶ Y) ≃ (Y ⟶ X)

Groupoid.inv is involutive.

Equations
  • CategoryTheory.Groupoid.invEquiv = { toFun := CategoryTheory.Groupoid.inv, invFun := CategoryTheory.Groupoid.inv, left_inv := ⋯, right_inv := ⋯ }
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@[instance 100]
instance CategoryTheory.groupoidHasInvolutiveReverse {C : Type u} [CategoryTheory.Groupoid C] :
Quiver.HasInvolutiveReverse C
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@[simp]
theorem CategoryTheory.Groupoid.reverse_eq_inv {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} (f : X ⟶ Y) :
Quiver.reverse f = CategoryTheory.Groupoid.inv f
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instance CategoryTheory.functorMapReverse {C : Type u} [CategoryTheory.Groupoid C] {D : Type u_1} [CategoryTheory.Groupoid D] (F : CategoryTheory.Functor C D) :
F.MapReverse
Equations
  • ⋯ = ⋯
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def CategoryTheory.Groupoid.isoEquivHom {C : Type u} [CategoryTheory.Groupoid C] (X : C) (Y : C) :
(X ≅ Y) ≃ (X ⟶ Y)

In a groupoid, isomorphisms are equivalent to morphisms.

Equations
  • One or more equations did not get rendered due to their size.
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@[simp]
theorem CategoryTheory.Groupoid.invFunctor_map (C : Type u) [CategoryTheory.Groupoid C] {X : C} {Y : C} (f : X ⟶ Y) :
(CategoryTheory.Groupoid.invFunctor C).map f = (CategoryTheory.Groupoid.inv f).op
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@[simp]
theorem CategoryTheory.Groupoid.invFunctor_obj (C : Type u) [CategoryTheory.Groupoid C] (unop : C) :
(CategoryTheory.Groupoid.invFunctor C).obj unop = Opposite.op unop
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noncomputable def CategoryTheory.Groupoid.invFunctor (C : Type u) [CategoryTheory.Groupoid C] :
CategoryTheory.Functor C Cᵒᵖ

The functor from a groupoid C to its opposite sending every morphism to its inverse.

Equations
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noncomputable def CategoryTheory.Groupoid.ofIsIso {C : Type u} [CategoryTheory.Category.{v, u} C] (all_is_iso : ∀ {X Y : C} (f : X ⟶ Y), CategoryTheory.IsIso f) :
CategoryTheory.Groupoid C

A category where every morphism IsIso is a groupoid.

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def CategoryTheory.Groupoid.ofHomUnique {C : Type u} [CategoryTheory.Category.{v, u} C] (all_unique : {X Y : C} → Unique (X ⟶ Y)) :
CategoryTheory.Groupoid C

A category with a unique morphism between any two objects is a groupoid

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instance CategoryTheory.InducedCategory.groupoid {C : Type u} (D : Type u₂) [CategoryTheory.Groupoid D] (F : C → D) :
CategoryTheory.Groupoid (CategoryTheory.InducedCategory D F)
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instance CategoryTheory.groupoidPi {I : Type u} {J : I → Type u₂} [(i : I) → CategoryTheory.Groupoid (J i)] :
CategoryTheory.Groupoid ((i : I) → J i)
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instance CategoryTheory.groupoidProd {α : Type u} {β : Type v} [CategoryTheory.Groupoid α] [CategoryTheory.Groupoid β] :
CategoryTheory.Groupoid (α × β)
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