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Mathlib.CategoryTheory.PUnit

The category Discrete PUnit #

We define star : C ⥤ Discrete PUnit sending everything to PUnit.star, show that any two functors to Discrete PUnit are naturally isomorphic, and construct the equivalence (Discrete PUnit ⥤ C) ≌ C.

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def CategoryTheory.Functor.star (C : Type u) [CategoryTheory.Category.{v, u} C] :
CategoryTheory.Functor C (CategoryTheory.Discrete PUnit.{w + 1})

The constant functor sending everything to PUnit.star.

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    @[simp]
    theorem CategoryTheory.Functor.star_map_down_down (C : Type u) [CategoryTheory.Category.{v, u} C] {X✝ Y✝ : C} (x✝ : X✝ ⟶ Y✝) :
    ⋯ = ⋯
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    @[simp]
    theorem CategoryTheory.Functor.star_obj_as (C : Type u) [CategoryTheory.Category.{v, u} C] (x✝ : C) :
    ((CategoryTheory.Functor.star C).obj x✝).as = PUnit.unit
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    def CategoryTheory.Functor.punitExt {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (CategoryTheory.Discrete PUnit.{w + 1})) :
    F ≅ G

    Any two functors to Discrete PUnit are isomorphic.

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      @[simp]
      theorem CategoryTheory.Functor.punitExt_hom_app_down_down {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (CategoryTheory.Discrete PUnit.{w + 1})) (X : C) :
      ⋯ = ⋯
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      @[simp]
      theorem CategoryTheory.Functor.punitExt_inv_app_down_down {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (CategoryTheory.Discrete PUnit.{w + 1})) (X : C) :
      ⋯ = ⋯
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      theorem CategoryTheory.Functor.punit_ext' {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor C (CategoryTheory.Discrete PUnit.{w + 1})) :
      F = G

      Any two functors to Discrete PUnit are equal. You probably want to use punitExt instead of this.

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      @[reducible, inline]
      abbrev CategoryTheory.Functor.fromPUnit {C : Type u} [CategoryTheory.Category.{v, u} C] (X : C) :
      CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C

      The functor from Discrete PUnit sending everything to the given object.

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        def CategoryTheory.Functor.equiv {C : Type u} [CategoryTheory.Category.{v, u} C] :
        CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C ≌ C

        Functors from Discrete PUnit are equivalent to the category itself.

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          @[simp]
          theorem CategoryTheory.Functor.equiv_functor_obj {C : Type u} [CategoryTheory.Category.{v, u} C] (F : CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C) :
          CategoryTheory.Functor.equiv.functor.obj F = F.obj { as := PUnit.unit }
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          @[simp]
          theorem CategoryTheory.Functor.equiv_inverse {C : Type u} [CategoryTheory.Category.{v, u} C] :
          CategoryTheory.Functor.equiv.inverse = CategoryTheory.Functor.const (CategoryTheory.Discrete PUnit.{w + 1})
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          @[simp]
          theorem CategoryTheory.Functor.equiv_counitIso {C : Type u} [CategoryTheory.Category.{v, u} C] :
          CategoryTheory.Functor.equiv.counitIso = CategoryTheory.NatIso.ofComponents CategoryTheory.Iso.refl ⋯
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          @[simp]
          theorem CategoryTheory.Functor.equiv_functor_map {C : Type u} [CategoryTheory.Category.{v, u} C] {X✝ Y✝ : CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C} (θ : X✝ ⟶ Y✝) :
          CategoryTheory.Functor.equiv.functor.map θ = θ.app { as := PUnit.unit }
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          @[simp]
          theorem CategoryTheory.Functor.equiv_unitIso {C : Type u} [CategoryTheory.Category.{v, u} C] :
          CategoryTheory.Functor.equiv.unitIso = CategoryTheory.NatIso.ofComponents (fun (x : CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C) => CategoryTheory.Discrete.natIso fun (x_1 : CategoryTheory.Discrete PUnit.{w + 1}) => CategoryTheory.Iso.refl (((CategoryTheory.Functor.id (CategoryTheory.Functor (CategoryTheory.Discrete PUnit.{w + 1}) C)).obj x).obj x_1)) ⋯
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          theorem CategoryTheory.equiv_punit_iff_unique (C : Type u) [CategoryTheory.Category.{v, u} C] :
          Nonempty (C ≌ CategoryTheory.Discrete PUnit.{w + 1}) ↔ Nonempty C ∧ ∀ (x y : C), Nonempty (Unique (x ⟶ y))

          A category being equivalent to PUnit is equivalent to it having a unique morphism between any two objects. (In fact, such a category is also a groupoid; see CategoryTheory.Groupoid.ofHomUnique)