# Documentation

Mathlib.Combinatorics.Quiver.Subquiver

## Wide subquivers #

A wide subquiver H of a quiver H consists of a subset of the edge set a ⟶ b⟶ b for every pair of vertices a b : V. We include 'wide' in the name to emphasize that these subquivers by definition contain all vertices.

def WideSubquiver (V : Type u_1) [inst : ] :
Type (maxu_1v)

A wide subquiver H of G picks out a set H a b of arrows from a to b for every pair of vertices a b.

NB: this does not work for Prop-valued quivers. It requires G : Quiver.{v+1} V.

Equations
def WideSubquiver.toType (V : Type u) [inst : ] :

A type synonym for V, when thought of as a quiver having only the arrows from some WideSubquiver.

Equations
instance wideSubquiverHasCoeToSort {V : Type u} [inst : ] :
CoeSort () (Type u)
Equations
• wideSubquiverHasCoeToSort = { coe := fun H => }
instance WideSubquiver.quiver {V : Type u_1} [inst : ] (H : ) :

A wide subquiver viewed as a quiver on its own.

Equations
• = { Hom := fun a b => { f // f H a b } }
instance Quiver.instBotWideSubquiver {V : Type u_1} [inst : ] :
Bot ()
Equations
• Quiver.instBotWideSubquiver = { bot := fun x x_1 => }
instance Quiver.instTopWideSubquiver {V : Type u_1} [inst : ] :
Top ()
Equations
• Quiver.instTopWideSubquiver = { top := fun x x_1 => Set.univ }
noncomputable instance Quiver.instInhabitedWideSubquiver {V : Type u_1} [inst : ] :
Equations
• Quiver.instInhabitedWideSubquiver = { default := }
theorem Quiver.Total.ext {V : Type u} :
∀ {inst : } (x y : ), x.left = y.leftx.right = y.rightHEq x.hom y.homx = y
theorem Quiver.Total.ext_iff {V : Type u} :
∀ {inst : } (x y : ), x = y x.left = y.left x.right = y.right HEq x.hom y.hom
structure Quiver.Total (V : Type u) [inst : ] :
Sort (max(u+1)v)
• the source vertex of an arrow

left : V
• the target vertex of an arrow

right : V
• an arrow

hom : left right

Total V is the type of all arrows of V.

Instances For
def Quiver.wideSubquiverEquivSetTotal {V : Type u_1} [inst : ] :

A wide subquiver of G can equivalently be viewed as a total set of arrows.

Equations
• One or more equations did not get rendered due to their size.
def Quiver.Labelling (V : Type u) [inst : ] (L : Sort u_2) :
Sort (imax(u+1)(u+1)u_1u_2)

An L-labelling of a quiver assigns to every arrow an element of L.

Equations
• = (a b : V⦄ → (a b) → L)
instance Quiver.instInhabitedLabelling {V : Type u} [inst : ] (L : Sort u_2) [inst : ] :
Equations
• = { default := fun x x_1 x => default }