# Documentation

Mathlib.Combinatorics.Quiver.Basic

# 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⟶ b of arrows from a to b. This is a very permissive notion of directed graph.

## Implementation notes #

Currently Quiver is defined with arrow : V → V → Sort v→ V → Sort 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 (maxuv)
• The type of edges/arrows/morphisms between a given source and target.

Hom : VVSort v

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

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

Instances

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

Equations
structure Prefunctor (V : Type u₁) [inst : ] (W : Type u₂) [inst : ] :
Sort (max(max(max(u₁+1)(u₂+1))v₁)v₂)
• The action of a (pre)functor on vertices/objects.

obj : VW
• The action of a (pre)functor on edges/arrows/morphisms.

map : {X Y : V} → (X Y) → (obj X obj Y)

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

Instances For
theorem Prefunctor.ext {V : Type u} [inst : ] {W : Type u₂} [inst : ] {F : V ⥤q W} {G : V ⥤q W} (h_obj : ∀ (X : V), = ) (h_map : ∀ (X Y : V) (f : X Y), = Eq.recOn (_ : = ) (Eq.recOn (_ : = ) ())) :
F = G
@[simp]
theorem Prefunctor.id_obj (V : Type u_1) [inst : ] (X : V) :
@[simp]
theorem Prefunctor.id_map (V : Type u_1) [inst : ] :
∀ {X Y : V} (f : X Y), Prefunctor.map (𝟭q V) f = f
def Prefunctor.id (V : Type u_1) [inst : ] :
V ⥤q V

The identity morphism between quivers.

Equations
• 𝟭q V = { obj := fun X => X, map := fun {X Y} f => f }
instance Prefunctor.instInhabitedPrefunctor (V : Type u_1) [inst : ] :
Equations
@[simp]
theorem Prefunctor.comp_obj {U : Type u_1} [inst : ] {V : Type u_3} [inst : ] {W : Type u_5} [inst : ] (F : U ⥤q V) (G : V ⥤q W) (X : U) :
@[simp]
theorem Prefunctor.comp_map {U : Type u_1} [inst : ] {V : Type u_3} [inst : ] {W : Type u_5} [inst : ] (F : U ⥤q V) (G : V ⥤q W) :
∀ {X Y : U} (f : X Y), Prefunctor.map (F ⋙q G) f = Prefunctor.map G ()
def Prefunctor.comp {U : Type u_1} [inst : ] {V : Type u_3} [inst : ] {W : Type u_5} [inst : ] (F : U ⥤q V) (G : V ⥤q W) :
U ⥤q W

Composition of morphisms between quivers.

Equations
@[simp]
theorem Prefunctor.comp_id {U : Type u_1} {V : Type u_2} [inst : ] [inst : ] (F : U ⥤q V) :
F ⋙q 𝟭q V = F
@[simp]
theorem Prefunctor.id_comp {U : Type u_1} {V : Type u_2} [inst : ] [inst : ] (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} [inst : ] [inst : ] [inst : ] [inst : ] (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.

Equations

Notation for composition of prefunctors.

Equations

Notation for the identity prefunctor on a quiver.

Equations
instance Quiver.opposite {V : Type u_1} [inst : ] :

Vᵒᵖ reverses the direction of all arrows of V.

Equations
• Quiver.opposite = { Hom := fun a b => }
def Quiver.Hom.op {V : Type u_1} [inst : ] {X : V} {Y : V} (f : X Y) :

The opposite of an arrow in V.

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

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

Equations
def Quiver.Empty (V : Type u) :

A type synonym for a quiver with no arrows.

Equations
instance Quiver.emptyQuiver (V : Type u) :
Equations
@[simp]
theorem Quiver.empty_arrow {V : Type u} (a : ) (b : ) :
(a b) = PEmpty
def Quiver.IsThin (V : Type u) [inst : ] :

A quiver is thin if it has no parallel arrows.

Equations