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Mathlib.CategoryTheory.Comma.Over.StrictInitial

Over X when C has strict initial objects #

In this file we define the canonical equivalence of Over X with Discrete PUnit when C has strict initial objects. We also provide the variants for P.Over Q X and the dual versions.

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noncomputable def CategoryTheory.overEquivOfIsInitial {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
Over X ≌ Discrete PUnit.{w + 1}

If C has strict initial objects and X is an initial object, the category Over X is equivalent to a point.

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    @[simp]
    theorem CategoryTheory.overEquivOfIsInitial_counitIso {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
    (overEquivOfIsInitial.{w, v_1, u_1} X h).counitIso = Iso.refl ((Functor.fromPUnit (Over.mk (CategoryStruct.id X))).comp (Functor.star (Over X)))
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    @[simp]
    theorem CategoryTheory.overEquivOfIsInitial_functor {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
    (overEquivOfIsInitial.{w, v_1, u_1} X h).functor = Functor.star (Over X)
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    @[simp]
    theorem CategoryTheory.overEquivOfIsInitial_inverse {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
    (overEquivOfIsInitial.{w, v_1, u_1} X h).inverse = Functor.fromPUnit (Over.mk (CategoryStruct.id X))
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    @[simp]
    theorem CategoryTheory.overEquivOfIsInitial_unitIso {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
    (overEquivOfIsInitial.{w, v_1, u_1} X h).unitIso = NatIso.ofComponents (fun (A : Over X) => Over.isoMk (asIso A.hom) ⋯) ⋯
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    noncomputable def CategoryTheory.underEquivOfIsTerminal {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
    Under X ≌ Discrete PUnit.{w + 1}

    If C has strict terminal objects and X is a terminal object, the category Under X is equivalent to a point.

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      @[simp]
      theorem CategoryTheory.underEquivOfIsTerminal_inverse {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
      (underEquivOfIsTerminal.{w, v_1, u_1} X h).inverse = Functor.fromPUnit (Under.mk (CategoryStruct.id X))
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      @[simp]
      theorem CategoryTheory.underEquivOfIsTerminal_unitIso {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
      (underEquivOfIsTerminal.{w, v_1, u_1} X h).unitIso = NatIso.ofComponents (fun (A : Under X) => Under.isoMk (asIso A.hom).symm ⋯) ⋯
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      @[simp]
      theorem CategoryTheory.underEquivOfIsTerminal_counitIso {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
      (underEquivOfIsTerminal.{w, v_1, u_1} X h).counitIso = Iso.refl ((Functor.fromPUnit (Under.mk (CategoryStruct.id X))).comp (Functor.star (Under X)))
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      @[simp]
      theorem CategoryTheory.underEquivOfIsTerminal_functor {C : Type u_1} [Category.{v_1, u_1} C] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
      (underEquivOfIsTerminal.{w, v_1, u_1} X h).functor = Functor.star (Under X)
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      noncomputable def CategoryTheory.MorphismProperty.overEquivOfIsInitial {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
      P.Over Q X ≌ Discrete PUnit.{w + 1}

      If C has strict initial objects and X is an initial object, the category P.Over Q X is equivalent to a point.

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        @[simp]
        theorem CategoryTheory.MorphismProperty.overEquivOfIsInitial_counitIso {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
        (overEquivOfIsInitial.{w, v_1, u_1} P Q X h).counitIso = Iso.refl ((Functor.fromPUnit (Over.mk Q (CategoryStruct.id X) ⋯)).comp (Functor.star (P.Over Q X)))
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        @[simp]
        theorem CategoryTheory.MorphismProperty.overEquivOfIsInitial_unitIso {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
        (overEquivOfIsInitial.{w, v_1, u_1} P Q X h).unitIso = NatIso.ofComponents (fun (A : P.Over Q X) => Over.isoMk (asIso A.hom) ⋯) ⋯
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        @[simp]
        theorem CategoryTheory.MorphismProperty.overEquivOfIsInitial_functor {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
        (overEquivOfIsInitial.{w, v_1, u_1} P Q X h).functor = Functor.star (P.Over Q X)
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        @[simp]
        theorem CategoryTheory.MorphismProperty.overEquivOfIsInitial_inverse {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictInitialObjects C] (X : C) (h : Limits.IsInitial X) :
        (overEquivOfIsInitial.{w, v_1, u_1} P Q X h).inverse = Functor.fromPUnit (Over.mk Q (CategoryStruct.id X) ⋯)
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        noncomputable def CategoryTheory.MorphismProperty.underEquivOfIsTerminal {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
        P.Under Q X ≌ Discrete PUnit.{w + 1}

        If C has strict terminal objects and X is a terminal object, the category P.Under Q X is equivalent to a point.

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          @[simp]
          theorem CategoryTheory.MorphismProperty.underEquivOfIsTerminal_unitIso {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
          (underEquivOfIsTerminal.{w, v_1, u_1} P Q X h).unitIso = NatIso.ofComponents (fun (A : P.Under Q X) => Under.isoMk (asIso A.hom).symm ⋯) ⋯
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          @[simp]
          theorem CategoryTheory.MorphismProperty.underEquivOfIsTerminal_counitIso {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
          (underEquivOfIsTerminal.{w, v_1, u_1} P Q X h).counitIso = Iso.refl ((Functor.fromPUnit (Under.mk Q (CategoryStruct.id X) ⋯)).comp (Functor.star (P.Under Q X)))
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          @[simp]
          theorem CategoryTheory.MorphismProperty.underEquivOfIsTerminal_inverse {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
          (underEquivOfIsTerminal.{w, v_1, u_1} P Q X h).inverse = Functor.fromPUnit (Under.mk Q (CategoryStruct.id X) ⋯)
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          @[simp]
          theorem CategoryTheory.MorphismProperty.underEquivOfIsTerminal_functor {C : Type u_1} [Category.{v_1, u_1} C] (P Q : MorphismProperty C) [P.ContainsIdentities] [Q.IsMultiplicative] [Q.RespectsIso] [Limits.HasStrictTerminalObjects C] (X : C) (h : Limits.IsTerminal X) :
          (underEquivOfIsTerminal.{w, v_1, u_1} P Q X h).functor = Functor.star (P.Under Q X)