Quivers #

This module defines quivers. A quiver on a type V of vertices assigns to every pair a b : V of vertices a type a ⟶ b of arrows from a to b. This is a very permissive notion of directed graph.

Implementation notes #

Currently Quiver is defined with Hom : V → V → Sort v. This is different from the category theory setup, where we insist that morphisms live in some Type. There's some balance here: it's nice to allow Prop to ensure there are no multiple arrows, but it is also results in error-prone universe signatures when constraints require a Type.

class Quiver (V : Type u) :
Type (max u v)

A quiver G on a type V of vertices assigns to every pair a b : V of vertices a type a ⟶ b of arrows from a to b.

For graphs with no repeated edges, one can use Quiver.{0} V, which ensures a ⟶ b : Prop. For multigraphs, one can use Quiver.{v+1} V, which ensures a ⟶ b : Type v.

Because Category will later extend this class, we call the field Hom. Except when constructing instances, you should rarely see this, and use the ⟶ notation instead.

• Hom : VVSort v

The type of edges/arrows/morphisms between a given source and target.

Instances

Notation for the type of edges/arrows/morphisms between a given source and target in a quiver or category.

Equations
Instances For
structure Prefunctor (V : Type u₁) [] (W : Type u₂) [] :
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 : VW

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
theorem Prefunctor.mk_obj {V : Type u_1} {W : Type u_2} [] [] {obj : VW} {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} [] [] {obj : VW} {map : {X Y : V} → (X Y)(obj X obj Y)} {X : V} {Y : V} {f : X Y} :
{ obj := obj, map := map }.map f = map f
theorem Prefunctor.ext {V : Type u} [] {W : Type u₂} [] {F : V ⥤q W} {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
@[simp]
theorem Prefunctor.id_obj (V : Type u_1) [] (X : V) :
(𝟭q V).obj X = X
@[simp]
theorem Prefunctor.id_map (V : Type u_1) [] :
∀ {X Y : V} (f : X Y), (𝟭q V).map f = f
def Prefunctor.id (V : Type u_1) [] :
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 }
Instances For
instance Prefunctor.instInhabited (V : Type u_1) [] :
Equations
@[simp]
theorem Prefunctor.comp_map {U : Type u_1} [] {V : Type u_2} [] {W : Type u_3} [] (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} [] {V : Type u_2} [] {W : Type u_3} [] (F : U ⥤q V) (G : V ⥤q W) (X : U) :
(F ⋙q G).obj X = G.obj (F.obj X)
def Prefunctor.comp {U : Type u_1} [] {V : Type u_2} [] {W : Type u_3} [] (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) }
Instances For
@[simp]
theorem Prefunctor.comp_id {U : Type u_1} {V : Type u_2} [] [] (F : U ⥤q V) :
F ⋙q 𝟭q V = F
@[simp]
theorem Prefunctor.id_comp {U : Type u_1} {V : Type u_2} [] [] (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} [] [] [] [] (F : U ⥤q V) (G : V ⥤q W) (H : W ⥤q Z) :
F ⋙q G ⋙q H = F ⋙q (G ⋙q H)

Notation for a prefunctor between quivers.

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Notation for composition of prefunctors.

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Notation for the identity prefunctor on a quiver.

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theorem Prefunctor.congr_map {U : Type u_1} {V : Type u_2} [] [] (F : U ⥤q V) {X : U} {Y : U} {f : X Y} {g : X Y} (h : f = g) :
F.map f = F.map g
instance Quiver.opposite {V : Type u_1} [] :

Vᵒᵖ reverses the direction of all arrows of V.

Equations
def Quiver.Hom.op {V : Type u_1} [] {X : V} {Y : V} (f : X Y) :

The opposite of an arrow in V.

Equations
• f.op =
Instances For
def Quiver.Hom.unop {V : Type u_1} [] {X : Vᵒᵖ} {Y : Vᵒᵖ} (f : X Y) :

Given an arrow in Vᵒᵖ, we can take the "unopposite" back in V.

Equations
• f.unop =
Instances For
def Quiver.Empty (V : Type u) :

A type synonym for a quiver with no arrows.

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Instances For
instance Quiver.emptyQuiver (V : Type u) :
Equations
@[simp]
theorem Quiver.empty_arrow {V : Type u} (a : ) (b : ) :
@[reducible, inline]
abbrev Quiver.IsThin (V : Type u) [] :

A quiver is thin if it has no parallel arrows.

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