Documentation

Mathlib.CategoryTheory.Category.Cat.Limit

The category of small categories has all small limits. #

An object in the limit consists of a family of objects, which are carried to one another by the functors in the diagram. A morphism between two such objects is a family of morphisms between the corresponding objects, which are carried to one another by the action on morphisms of the functors in the diagram.

Future work #

Can the indexing category live in a lower universe?

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instance CategoryTheory.Cat.HasLimits.categoryObjects {J : Type v} [CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CategoryTheory.Cat} {j : J} :
CategoryTheory.SmallCategory ((F.comp CategoryTheory.Cat.objects).obj j)
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  • CategoryTheory.Cat.HasLimits.categoryObjects = (F.obj j).str
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def CategoryTheory.Cat.HasLimits.homDiagram {J : Type v} [CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CategoryTheory.Cat} (X Y : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) :
CategoryTheory.Functor J (Type v)

Auxiliary definition: the diagram whose limit gives the morphism space between two objects of the limit category.

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    @[simp]
    theorem CategoryTheory.Cat.HasLimits.homDiagram_map {J : Type v} [CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CategoryTheory.Cat} (X Y : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) {X✝ Y✝ : J} (f : X✝ ⟶ Y✝) (g : CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) X✝ X ⟶ CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) X✝ Y) :
    (CategoryTheory.Cat.HasLimits.homDiagram X Y).map f g = CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯) (CategoryTheory.CategoryStruct.comp ((F.map f).map g) (CategoryTheory.eqToHom ⋯))
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    @[simp]
    theorem CategoryTheory.Cat.HasLimits.homDiagram_obj {J : Type v} [CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CategoryTheory.Cat} (X Y : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) (j : J) :
    (CategoryTheory.Cat.HasLimits.homDiagram X Y).obj j = (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) j X ⟶ CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) j Y)
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    instance CategoryTheory.Cat.HasLimits.instCategoryLimitCompObjects {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
    CategoryTheory.Category.{v, v} (CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects))
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    @[simp]
    theorem CategoryTheory.Cat.HasLimits.instCategoryLimitCompObjects_id {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (X : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) :
    CategoryTheory.CategoryStruct.id X = CategoryTheory.Limits.Types.Limit.mk (CategoryTheory.Cat.HasLimits.homDiagram X X) (fun (x : J) => CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) x X)) ⋯
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    @[simp]
    theorem CategoryTheory.Cat.HasLimits.instCategoryLimitCompObjects_comp {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) {X Y Z : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)} (f : X ⟶ Y) (g : Y ⟶ Z) :
    CategoryTheory.CategoryStruct.comp f g = CategoryTheory.Limits.Types.Limit.mk (CategoryTheory.Cat.HasLimits.homDiagram X Z) (fun (j : J) => CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.limit.π (CategoryTheory.Cat.HasLimits.homDiagram X Y) j f) (CategoryTheory.Limits.limit.π (CategoryTheory.Cat.HasLimits.homDiagram Y Z) j g)) ⋯
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    def CategoryTheory.Cat.HasLimits.limitConeX {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
    CategoryTheory.Cat

    Auxiliary definition: the limit category.

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      @[simp]
      theorem CategoryTheory.Cat.HasLimits.limitConeX_α {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
      ↑(CategoryTheory.Cat.HasLimits.limitConeX F) = CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)
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      @[simp]
      theorem CategoryTheory.Cat.HasLimits.limitConeX_str {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
      (CategoryTheory.Cat.HasLimits.limitConeX F).str = inferInstance
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      def CategoryTheory.Cat.HasLimits.limitCone {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
      CategoryTheory.Limits.Cone F

      Auxiliary definition: the cone over the limit category.

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      • One or more equations did not get rendered due to their size.
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        @[simp]
        theorem CategoryTheory.Cat.HasLimits.limitCone_π_app_obj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (j : J) (a✝ : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) :
        ((CategoryTheory.Cat.HasLimits.limitCone F).π.app j).obj a✝ = CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) j a✝
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        @[simp]
        theorem CategoryTheory.Cat.HasLimits.limitCone_π_app_map {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (j : J) {X✝ Y✝ : ↑(((CategoryTheory.Functor.const J).obj (CategoryTheory.Cat.HasLimits.limitConeX F)).obj j)} (f : X✝ ⟶ Y✝) :
        ((CategoryTheory.Cat.HasLimits.limitCone F).π.app j).map f = CategoryTheory.Limits.limit.π (CategoryTheory.Cat.HasLimits.homDiagram X✝ Y✝) j f
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        @[simp]
        theorem CategoryTheory.Cat.HasLimits.limitCone_pt {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
        (CategoryTheory.Cat.HasLimits.limitCone F).pt = CategoryTheory.Cat.HasLimits.limitConeX F
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        def CategoryTheory.Cat.HasLimits.limitConeLift {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (s : CategoryTheory.Limits.Cone F) :
        s.pt ⟶ CategoryTheory.Cat.HasLimits.limitConeX F

        Auxiliary definition: the universal morphism to the proposed limit cone.

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          @[simp]
          theorem CategoryTheory.Cat.HasLimits.limitConeLift_obj {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (s : CategoryTheory.Limits.Cone F) (a✝ : { pt := ↑s.pt, π := { app := fun (j : J) => (s.π.app j).obj, naturality := ⋯ } }.pt) :
          (CategoryTheory.Cat.HasLimits.limitConeLift F s).obj a✝ = CategoryTheory.Limits.limit.lift (F.comp CategoryTheory.Cat.objects) { pt := ↑s.pt, π := { app := fun (j : J) => (s.π.app j).obj, naturality := ⋯ } } a✝
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          @[simp]
          theorem CategoryTheory.Cat.HasLimits.limitConeLift_map {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) (s : CategoryTheory.Limits.Cone F) {X✝ Y✝ : ↑s.pt} (f : X✝ ⟶ Y✝) :
          (CategoryTheory.Cat.HasLimits.limitConeLift F s).map f = CategoryTheory.Limits.Types.Limit.mk (CategoryTheory.Cat.HasLimits.homDiagram (CategoryTheory.Limits.limit.lift (F.comp CategoryTheory.Cat.objects) { pt := ↑s.pt, π := { app := fun (j : J) => (s.π.app j).obj, naturality := ⋯ } } X✝) (CategoryTheory.Limits.limit.lift (F.comp CategoryTheory.Cat.objects) { pt := ↑s.pt, π := { app := fun (j : J) => (s.π.app j).obj, naturality := ⋯ } } Y✝)) (fun (j : J) => CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯) (CategoryTheory.CategoryStruct.comp ((s.π.app j).map f) (CategoryTheory.eqToHom ⋯))) ⋯
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          @[simp]
          theorem CategoryTheory.Cat.HasLimits.limit_π_homDiagram_eqToHom {J : Type v} [CategoryTheory.SmallCategory J] {F : CategoryTheory.Functor J CategoryTheory.Cat} (X Y : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)) (j : J) (h : X = Y) :
          CategoryTheory.Limits.limit.π (CategoryTheory.Cat.HasLimits.homDiagram X Y) j (CategoryTheory.eqToHom h) = CategoryTheory.eqToHom ⋯
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          def CategoryTheory.Cat.HasLimits.limitConeIsLimit {J : Type v} [CategoryTheory.SmallCategory J] (F : CategoryTheory.Functor J CategoryTheory.Cat) :
          CategoryTheory.Limits.IsLimit (CategoryTheory.Cat.HasLimits.limitCone F)

          Auxiliary definition: the proposed cone is a limit cone.

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          Instances For
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            instance CategoryTheory.Cat.instHasLimits :
            CategoryTheory.Limits.HasLimits CategoryTheory.Cat

            The category of small categories has all small limits.

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            instance CategoryTheory.Cat.instPreservesLimitsObjects :
            CategoryTheory.Limits.PreservesLimits CategoryTheory.Cat.objects
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