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Mathlib.CategoryTheory.MorphismProperty.Limits

Relation of morphism properties with limits #

The following predicates are introduces for morphism properties P:

We define P.universally for the class of morphisms which satisfy P after any base change.

We also introduce properties IsStableUnderProductsOfShape, IsStableUnderLimitsOfShape, IsStableUnderFiniteProducts, and similar properties for colimits and coproducts.

Given a class of morphisms P, this is the class of pullbacks of morphisms in P.

Equations
theorem CategoryTheory.MorphismProperty.pullbacks_mk {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) {A B X Y : C} {f : A X} {q : A B} {p : X Y} {g : B Y} (sq : IsPullback f q p g) (hp : P p) :

Given a class of morphisms P, this is the class of pushouts of morphisms in P.

Equations
theorem CategoryTheory.MorphismProperty.pushouts_mk {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) {A B X Y : C} {f : A X} {q : A B} {p : X Y} {g : B Y} (sq : IsPushout f q p g) (hq : P q) :
theorem CategoryTheory.MorphismProperty.isomorphisms_le_pushouts {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) (h : ∀ (X : C), ∃ (A : C) (B : C) (p : A B) (_ : P p) (x : B X), IsIso p) :

If P : MorphismPropety C is such that any object in C maps to the target of some morphism in P, then P.pushouts contains the isomorphisms.

A morphism property is IsStableUnderBaseChange if the base change of such a morphism still falls in the class.

  • of_isPullback {X Y Y' S : C} {f : X S} {g : Y S} {f' : Y' Y} {g' : Y' X} (sq : IsPullback f' g' g f) (hg : P g) : P g'
Instances

    A morphism property is IsStableUnderCobaseChange if the cobase change of such a morphism still falls in the class.

    • of_isPushout {A A' B B' : C} {f : A A'} {g : A B} {f' : B B'} {g' : A' B'} (sq : IsPushout g f f' g') (hf : P f) : P f'
    Instances
      theorem CategoryTheory.MorphismProperty.of_isPullback {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [P.IsStableUnderBaseChange] {X Y Y' S : C} {f : X S} {g : Y S} {f' : Y' Y} {g' : Y' X} (sq : IsPullback f' g' g f) (hg : P g) :
      P g'
      theorem CategoryTheory.MorphismProperty.IsStableUnderBaseChange.mk' {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [P.RespectsIso] (hP₂ : ∀ (X Y S : C) (f : X S) (g : Y S) [inst : Limits.HasPullback f g], P gP (Limits.pullback.fst f g)) :

      Alternative constructor for IsStableUnderBaseChange.

      @[deprecated CategoryTheory.MorphismProperty.pullback_fst (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.pullback_fst.

      @[deprecated CategoryTheory.MorphismProperty.pullback_snd (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.pullback_snd.

      @[deprecated CategoryTheory.MorphismProperty.baseChange_obj (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.baseChange_obj.

      theorem CategoryTheory.MorphismProperty.baseChange_map {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [Limits.HasPullbacks C] [P.IsStableUnderBaseChange] {S S' : C} (f : S' S) {X Y : Over S} (g : X Y) (H : P g.left) :
      @[deprecated CategoryTheory.MorphismProperty.baseChange_map (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.baseChange_map.

      theorem CategoryTheory.MorphismProperty.pullback_map {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [Limits.HasPullbacks C] [P.IsStableUnderBaseChange] [P.IsStableUnderComposition] {S X X' Y Y' : C} {f : X S} {g : Y S} {f' : X' S} {g' : Y' S} {i₁ : X X'} {i₂ : Y Y'} (h₁ : P i₁) (h₂ : P i₂) (e₁ : f = CategoryStruct.comp i₁ f') (e₂ : g = CategoryStruct.comp i₂ g') :
      P (Limits.pullback.map f g f' g' i₁ i₂ (CategoryStruct.id S) )
      @[deprecated CategoryTheory.MorphismProperty.pullback_map (since := "2024-11-06")]
      theorem CategoryTheory.MorphismProperty.IsStableUnderBaseChange.pullback_map {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [Limits.HasPullbacks C] [P.IsStableUnderBaseChange] [P.IsStableUnderComposition] {S X X' Y Y' : C} {f : X S} {g : Y S} {f' : X' S} {g' : Y' S} {i₁ : X X'} {i₂ : Y Y'} (h₁ : P i₁) (h₂ : P i₂) (e₁ : f = CategoryStruct.comp i₁ f') (e₂ : g = CategoryStruct.comp i₂ g') :
      P (Limits.pullback.map f g f' g' i₁ i₂ (CategoryStruct.id S) )

      Alias of CategoryTheory.MorphismProperty.pullback_map.

      theorem CategoryTheory.MorphismProperty.of_isPushout {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [P.IsStableUnderCobaseChange] {A A' B B' : C} {f : A A'} {g : A B} {f' : B B'} {g' : A' B'} (sq : IsPushout g f f' g') (hf : P f) :
      P f'
      theorem CategoryTheory.MorphismProperty.IsStableUnderCobaseChange.mk' {C : Type u} [Category.{v, u} C] {P : MorphismProperty C} [P.RespectsIso] (hP₂ : ∀ (A B A' : C) (f : A A') (g : A B) [inst : Limits.HasPushout f g], P fP (Limits.pushout.inr f g)) :

      An alternative constructor for IsStableUnderCobaseChange.

      @[deprecated CategoryTheory.MorphismProperty.pushout_inl (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.pushout_inl.

      @[deprecated CategoryTheory.MorphismProperty.pushout_inr (since := "2024-11-06")]

      Alias of CategoryTheory.MorphismProperty.pushout_inr.

      The class of morphisms in C that are limits of shape J of natural transformations involving morphisms in W.

      theorem CategoryTheory.MorphismProperty.limitsOfShape.mk' {C : Type u} [Category.{v, u} C] {W : MorphismProperty C} {J : Type u_1} [Category.{u_2, u_1} J] (X₁ X₂ : Functor J C) (c₁ : Limits.Cone X₁) (c₂ : Limits.Cone X₂) (h₁ : Limits.IsLimit c₁) (h₂ : Limits.IsLimit c₂) (f : X₁ X₂) (hf : W.functorCategory J f) (φ : c₁.pt c₂.pt) ( : ∀ (j : J), CategoryStruct.comp φ (c₂.π.app j) = CategoryStruct.comp (c₁.π.app j) (f.app j)) :

      The property that a morphism property W is stable under limits indexed by a category J.

      Instances
        @[deprecated CategoryTheory.MorphismProperty.limitsOfShape_le (since := "2025-05-11")]

        Alias of CategoryTheory.MorphismProperty.limitsOfShape_le.

        @[deprecated CategoryTheory.MorphismProperty.limMap (since := "2025-05-11")]

        Alias of CategoryTheory.MorphismProperty.limMap.

        The class of morphisms in C that are colimits of shape J of natural transformations involving morphisms in W.

        theorem CategoryTheory.MorphismProperty.colimitsOfShape.mk' {C : Type u} [Category.{v, u} C] {W : MorphismProperty C} {J : Type u_1} [Category.{u_2, u_1} J] (X₁ X₂ : Functor J C) (c₁ : Limits.Cocone X₁) (c₂ : Limits.Cocone X₂) (h₁ : Limits.IsColimit c₁) (h₂ : Limits.IsColimit c₂) (f : X₁ X₂) (hf : W.functorCategory J f) (φ : c₁.pt c₂.pt) ( : ∀ (j : J), CategoryStruct.comp (c₁.ι.app j) φ = CategoryStruct.comp (f.app j) (c₂.ι.app j)) :

        The property that a morphism property W is stable under colimits indexed by a category J.

        Instances
          @[deprecated CategoryTheory.MorphismProperty.colimMap (since := "2025-05-11")]

          Alias of CategoryTheory.MorphismProperty.colimMap.

          @[deprecated CategoryTheory.MorphismProperty.colimitsOfShape_le (since := "2025-05-11")]

          Alias of CategoryTheory.MorphismProperty.colimitsOfShape_le.

          Given W : MorphismProperty C, this is class of morphisms that are isomorphic to a coproduct of a family (indexed by some J : Type w) of maps in W.

          Equations
          @[reducible, inline]

          The property that a morphism property W is stable under products indexed by a type J.

          Equations
          @[reducible, inline]

          The property that a morphism property W is stable under coproducts indexed by a type J.

          Equations
          theorem CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk {C : Type u} [Category.{v, u} C] (W : MorphismProperty C) (J : Type u_1) [W.RespectsIso] (hW : ∀ (X₁ X₂ : JC) [inst : Limits.HasProduct X₁] [inst_1 : Limits.HasProduct X₂] (f : (j : J) → X₁ j X₂ j), (∀ (j : J), W (f j))W (Limits.Pi.map f)) :
          theorem CategoryTheory.MorphismProperty.IsStableUnderCoproductsOfShape.mk {C : Type u} [Category.{v, u} C] (W : MorphismProperty C) (J : Type u_1) [W.RespectsIso] (hW : ∀ (X₁ X₂ : JC) [inst : Limits.HasCoproduct X₁] [inst_1 : Limits.HasCoproduct X₂] (f : (j : J) → X₁ j X₂ j), (∀ (j : J), W (f j))W (Limits.Sigma.map f)) :

          The condition that a property of morphisms is stable by finite products.

          Instances

            The condition that a property of morphisms is stable by finite coproducts.

            Instances
              @[deprecated CategoryTheory.MorphismProperty.IsStableUnderFiniteProducts.isStableUnderProductsOfShape "This is now an instance." (since := "2025-05-11")]

              Alias of CategoryTheory.MorphismProperty.IsStableUnderFiniteProducts.isStableUnderProductsOfShape.

              @[deprecated CategoryTheory.MorphismProperty.IsStableUnderFiniteCoproducts.isStableUnderCoproductsOfShape "This is now an instance." (since := "2025-05-11")]

              Alias of CategoryTheory.MorphismProperty.IsStableUnderFiniteCoproducts.isStableUnderCoproductsOfShape.

              The condition that a property of morphisms is stable by coproducts.

              Instances

                For P : MorphismProperty C, P.diagonal is a morphism property that holds for f : X ⟶ Y whenever P holds for X ⟶ Y xₓ Y.

                Equations

                If P is multiplicative and stable under base change, having the of-postcomp property wrt. Q is equivalent to Q implying P on the diagonal.

                P.universally holds for a morphism f : X ⟶ Y iff P holds for all X ×[Y] Y' ⟶ Y'.

                Equations
                theorem CategoryTheory.MorphismProperty.universally_mk' {C : Type u} [Category.{v, u} C] (P : MorphismProperty C) [P.RespectsIso] {X Y : C} (g : X Y) (H : ∀ {T : C} (f : T Y) [inst : Limits.HasPullback f g], P (Limits.pullback.fst f g)) :