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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    @[inline, reducible]
    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.

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

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

      Instances For
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        instance CategoryTheory.IsIso.of_groupoid {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} (f : X ⟶ Y) :
        CategoryTheory.IsIso f
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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_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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        @[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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        def CategoryTheory.Groupoid.invEquiv {C : Type u} [CategoryTheory.Groupoid C] {X : C} {Y : C} :
        (X ⟶ Y) ≃ (Y ⟶ X)

        Groupoid.inv is involutive.

        Instances For
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          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) :
          Prefunctor.MapReverse F.toPrefunctor
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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.

          Instances For
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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] (x : C) :
            (CategoryTheory.Groupoid.invFunctor C).obj x = Opposite.op x
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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.

            Instances For
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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.

              Instances For
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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

                Instances For
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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)