Documentation

Mathlib.Combinatorics.Quiver.Prefunctor

Morphisms of quivers #

source
structure Prefunctor (V : Type u₁) [Quiver V] (W : Type u₂) [Quiver W] :
Type (max (max (max u₁ u₂) v₁) v₂)

A morphism of quivers. As we will later have categorical functors extend this structure, we call it a Prefunctor.

  • obj : V → W

    The action of a (pre)functor on vertices/objects.

  • map {X Y : V} : (X ⟶ Y) → (self.obj X ⟶ self.obj Y)

    The action of a (pre)functor on edges/arrows/morphisms.

Instances For
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    theorem Prefunctor.mk_obj {V : Type u_1} {W : Type u_2} [Quiver V] [Quiver W] {obj : V → W} {map : {X Y : V} → (X ⟶ Y) → (obj X ⟶ obj Y)} {X : V} :
    { obj := obj, map := map }.obj X = obj X
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    theorem Prefunctor.mk_map {V : Type u_1} {W : Type u_2} [Quiver V] [Quiver W] {obj : V → W} {map : {X Y : V} → (X ⟶ Y) → (obj X ⟶ obj Y)} {X Y : V} {f : X ⟶ Y} :
    { obj := obj, map := map }.map f = map f
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    theorem Prefunctor.ext {V : Type u} [Quiver V] {W : Type u₂} [Quiver W] {F G : V ⥤q W} (h_obj : ∀ (X : V), F.obj X = G.obj X) (h_map : ∀ (X Y : V) (f : X ⟶ Y), F.map f = Eq.recOn ⋯ (Eq.recOn ⋯ (G.map f))) :
    F = G
    source
    theorem Prefunctor.ext' {V W : Type u} [Quiver V] [Quiver W] {F G : V ⥤q W} (h_obj : ∀ (X : V), F.obj X = G.obj X) (h_map : ∀ (X Y : V) (f : X ⟶ Y), F.map f = Quiver.homOfEq (G.map f) ⋯ ⋯) :
    F = G

    This may be a more useful form of Prefunctor.ext.

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    def Prefunctor.id (V : Type u_1) [Quiver V] :
    V ⥤q V

    The identity morphism between quivers.

    Instances For
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      @[simp]
      theorem Prefunctor.id_obj (V : Type u_1) [Quiver V] (X : V) :
      (𝟭q V).obj X = X
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      @[simp]
      theorem Prefunctor.id_map (V : Type u_1) [Quiver V] {X✝ Y✝ : V} (f : X✝ ⟶ Y✝) :
      (𝟭q V).map f = f
      source
      @[instance_reducible]
      instance Prefunctor.instInhabited (V : Type u_1) [Quiver V] :
      Inhabited (V ⥤q V)
      source
      def Prefunctor.comp {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] {W : Type u_3} [Quiver W] (F : U ⥤q V) (G : V ⥤q W) :
      U ⥤q W

      Composition of morphisms between quivers.

      Instances For
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        @[simp]
        theorem Prefunctor.comp_obj {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] {W : Type u_3} [Quiver W] (F : U ⥤q V) (G : V ⥤q W) (X : U) :
        (F ⋙q G).obj X = G.obj (F.obj X)
        source
        @[simp]
        theorem Prefunctor.comp_map {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] {W : Type u_3} [Quiver W] (F : U ⥤q V) (G : V ⥤q W) {X✝ Y✝ : U} (f : X✝ ⟶ Y✝) :
        (F ⋙q G).map f = G.map (F.map f)
        source
        @[simp]
        theorem Prefunctor.comp_id {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (F : U ⥤q V) :
        F ⋙q 𝟭q V = F
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        @[simp]
        theorem Prefunctor.id_comp {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (F : U ⥤q V) :
        𝟭q U ⋙q F = F
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        @[simp]
        theorem Prefunctor.comp_assoc {U : Type u_1} {V : Type u_2} {W : Type u_3} {Z : Type u_4} [Quiver U] [Quiver V] [Quiver W] [Quiver Z] (F : U ⥤q V) (G : V ⥤q W) (H : W ⥤q Z) :
        F ⋙q G ⋙q H = F ⋙q (G ⋙q H)
        source
        def Prefunctor.«term_⥤q_» :
        Lean.TrailingParserDescr

        Notation for a prefunctor between quivers.

        Instances For
          source
          def Prefunctor.«term_⋙q_» :
          Lean.TrailingParserDescr

          Notation for composition of prefunctors.

          Instances For
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            def Prefunctor.«term𝟭q» :
            Lean.ParserDescr

            Notation for the identity prefunctor on a quiver.

            Instances For
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              theorem Prefunctor.congr_map {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (F : U ⥤q V) {X Y : U} {f g : X ⟶ Y} (h : f = g) :
              F.map f = F.map g
              source
              theorem Prefunctor.congr_obj {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] {F G : U ⥤q V} (e : F = G) (X : U) :
              F.obj X = G.obj X

              An equality of prefunctors gives an equality on objects.

              source
              theorem Prefunctor.congr_hom {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] {F G : U ⥤q V} (e : F = G) {X Y : U} (f : X ⟶ Y) :
              Quiver.homOfEq (F.map f) ⋯ ⋯ = G.map f

              An equality of prefunctors gives an equality on homs.

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              @[simp]
              theorem Prefunctor.homOfEq_map {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (F : U ⥤q V) {X Y : U} (f : X ⟶ Y) {X' Y' : U} (hX : X = X') (hY : Y = Y') :
              F.map (Quiver.homOfEq f hX hY) = Quiver.homOfEq (F.map f) ⋯ ⋯

              Prefunctors commute with homOfEq.