Morphisms of quivers

structure Prefunctor (V : Type u₁) [Quiver V] (W : Type u₂) [Quiver W] : Sort (max (max (max (u₁ + 1) (u₂ + 1)) 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.

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

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

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

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.

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

The identity morphism between quivers.

Equations

• 𝟭q V = { obj := fun (X : V) => X, map := fun {X Y : V} (f : X ⟶ Y) => f }

@[simp]

theorem Prefunctor.id_obj (V : Type u_1) [Quiver V] (X : V) : (𝟭q V).obj X = X

@[simp]

theorem Prefunctor.id_map (V : Type u_1) [Quiver V] {X✝ Y✝ : V} (f : X✝ ⟶ Y✝) : (𝟭q V).map f = f

instance Prefunctor.instInhabited (V : Type u_1) [Quiver V] : Inhabited (V ⥤q V)

Equations

• Prefunctor.instInhabited V = { default := 𝟭q V }

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.

Equations

• F ⋙q G = { obj := fun (X : U) => G.obj (F.obj X), map := fun {X Y : U} (f : X ⟶ Y) => G.map (F.map f) }

@[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)

@[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)

@[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

@[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

@[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)

def Prefunctor.«term_⥤q_» : Lean.TrailingParserDescr

Notation for a prefunctor between quivers.

Equations

• Prefunctor.«term_⥤q_» = Lean.ParserDescr.trailingNode `Prefunctor.«term_⥤q_» 50 50 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⥤q ") (Lean.ParserDescr.cat `term 51))

def Prefunctor.«term_⋙q_» : Lean.TrailingParserDescr

Notation for composition of prefunctors.

Equations

• Prefunctor.«term_⋙q_» = Lean.ParserDescr.trailingNode `Prefunctor.«term_⋙q_» 60 60 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⋙q ") (Lean.ParserDescr.cat `term 61))

def Prefunctor.«term𝟭q» : Lean.ParserDescr

Notation for the identity prefunctor on a quiver.

Equations

• Prefunctor.«term𝟭q» = Lean.ParserDescr.node `Prefunctor.«term𝟭q» 1024 (Lean.ParserDescr.symbol "𝟭q")

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

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.

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.

@[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.