# Documentation

Mathlib.Combinatorics.Quiver.Symmetric

## Symmetric quivers and arrow reversal #

This file contains constructions related to symmetric quivers:

• Symmetrify V adds formal inverses to each arrow of V.
• HasReverse is the class of quivers where each arrow has an assigned formal inverse.
• HasInvolutiveReverse extends HasReverse by requiring that the reverse of the reverse is equal to the original arrow.
• Prefunctor.PreserveReverse is the class of prefunctors mapping reverses to reverses.
• Symmetrify.of, Symmetrify.lift, and the associated lemmas witness the universal property of Symmetrify.
def Quiver.Symmetrify (V : Type u_1) :
Type u_1

A type synonym for the symmetrized quiver (with an arrow both ways for each original arrow). NB: this does not work for Prop-valued quivers. It requires [Quiver.{v+1} V].

Equations
instance Quiver.symmetrifyQuiver (V : Type u) [inst : ] :
Equations
class Quiver.HasReverse (V : Type u_1) [inst : ] :
Type (maxu_1v)
• the map which sends an arrow to its reverse

reverse' : {a b : V} → (a b) → (b a)

A quiver HasReverse if we can reverse an arrow p from a to b to get an arrow p.reverse from b to a.

Instances
def Quiver.reverse {V : Type u_1} [inst : ] [inst : ] {a : V} {b : V} :
(a b) → (b a)

Reverse the direction of an arrow.

Equations
• Quiver.reverse = Quiver.HasReverse.reverse'
class Quiver.HasInvolutiveReverse (V : Type u_1) [inst : ] extends :
Type (maxu_1v)
• reverse is involutive

inv' : ∀ {a b : V} (f : a b),

A quiver HasInvolutiveReverse if reversing twice is the identity.

Instances
@[simp]
theorem Quiver.reverse_reverse {V : Type u_1} [inst : ] [h : ] {a : V} {b : V} (f : a b) :
@[simp]
theorem Quiver.reverse_inj {V : Type u_1} [inst : ] [h : ] {a : V} {b : V} (f : a b) (g : a b) :
f = g
theorem Quiver.eq_reverse_iff {V : Type u_1} [inst : ] [h : ] {a : V} {b : V} (f : a b) (g : b a) :
class Prefunctor.MapReverse {U : Type u_1} {V : Type u_2} [inst : ] [inst : ] [inst : ] [inst : ] (φ : U ⥤q V) :
• The image of a reverse is the reverse of the image.

map_reverse' : ∀ {u v : U} (e : u v),

A prefunctor preserving reversal of arrows

Instances
@[simp]
theorem Prefunctor.map_reverse {U : Type u_1} {V : Type u_2} [inst : ] [inst : ] [inst : ] [inst : ] (φ : U ⥤q V) [inst : ] {u : U} {v : U} (e : u v) :
instance Prefunctor.mapReverseComp {U : Type u_1} {V : Type u_2} {W : Type u_3} [inst : ] [inst : ] [inst : ] [inst : ] [inst : ] [inst : ] (φ : U ⥤q V) (ψ : V ⥤q W) [inst : ] [inst : ] :
Equations
• One or more equations did not get rendered due to their size.
instance Prefunctor.mapReverseId {U : Type u_1} [inst : ] [inst : ] :
Equations
Equations
• Quiver.instHasReverseSymmetrifySymmetrifyQuiver = { reverse' := fun {a b} e => }
Equations
@[simp]
theorem Quiver.symmetrify_reverse {V : Type u_1} [inst : ] {a : } {b : } (e : a b) :
@[inline]
abbrev Quiver.Hom.toPos {V : Type u_1} [inst : ] {X : V} {Y : V} (f : X Y) :
X Y

Shorthand for the "forward" arrow corresponding to f in symmetrify V

Equations
@[inline]
abbrev Quiver.Hom.toNeg {V : Type u_1} [inst : ] {X : V} {Y : V} (f : X Y) :
Y X

Shorthand for the "backward" arrow corresponding to f in symmetrify V

Equations
def Quiver.Path.reverse {V : Type u_1} [inst : ] [inst : ] {a : V} {b : V} :

Reverse the direction of a path.

Equations
@[simp]
theorem Quiver.Path.reverse_toPath {V : Type u_1} [inst : ] [inst : ] {a : V} {b : V} (f : a b) :
@[simp]
theorem Quiver.Path.reverse_comp {V : Type u_1} [inst : ] [inst : ] {a : V} {b : V} {c : V} (p : ) (q : ) :
@[simp]
theorem Quiver.Path.reverse_reverse {V : Type u_1} [inst : ] [h : ] {a : V} {b : V} (p : ) :
def Quiver.Symmetrify.of {V : Type u_1} [inst : ] :

The inclusion of a quiver in its symmetrification

Equations
• Quiver.Symmetrify.of = { obj := id, map := fun {X Y} => Sum.inl }
def Quiver.Symmetrify.lift {V : Type u_1} [inst : ] {V' : Type u_2} [inst : Quiver V'] [inst : ] (φ : V ⥤q V') :

Given a quiver V' with reversible arrows, a prefunctor to V' can be lifted to one from Symmetrify V to V'

Equations
• = { obj := φ.obj, map := fun {X Y} f => match f with | => | => }
theorem Quiver.Symmetrify.lift_spec {V : Type u_2} [inst : ] {V' : Type u_1} [inst : Quiver V'] [inst : ] (φ : V ⥤q V') :
Quiver.Symmetrify.of ⋙q = φ
theorem Quiver.Symmetrify.lift_reverse {V : Type u_2} [inst : ] {V' : Type u_1} [inst : Quiver V'] [h : ] (φ : V ⥤q V') {X : } {Y : } (f : X Y) :
theorem Quiver.Symmetrify.lift_unique {V : Type u_2} [inst : ] {V' : Type u_1} [inst : Quiver V'] [inst : ] (φ : V ⥤q V') (Φ : ) (hΦ : Quiver.Symmetrify.of ⋙q Φ = φ) (hΦinv : ∀ {X Y : } (f : X Y), ) :

lift φ is the only prefunctor extending φ and preserving reverses.

instance Quiver.Push.instHasReversePushInstQuiverPush {V : Type u_1} [inst : ] {V' : Type u_2} (σ : VV') [inst : ] :
Equations
• One or more equations did not get rendered due to their size.
instance Quiver.Push.instHasInvolutiveReversePushInstQuiverPush {V : Type u_1} [inst : ] {V' : Type u_2} (σ : VV') [h : ] :
Equations
theorem Quiver.Push.of_reverse {V : Type u_1} [inst : ] {V' : Type u_2} (σ : VV') [inst : ] (X : V) (Y : V) (f : X Y) :
instance Quiver.Push.ofMapReverse {V : Type u_1} [inst : ] {V' : Type u_2} (σ : VV') [h : ] :
Equations
• One or more equations did not get rendered due to their size.
def Quiver.IsPreconnected (V : Type u_1) [inst : ] :

A quiver is preconnected iff there exists a path between any pair of vertices. Note that if V doesn't HasReverse, then the definition is stronger than simply having a preconnected underlying simple_graph`, since a path in one direction doesn't induce one in the other.

Equations