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Mathlib.Combinatorics.Quiver.Symmetric

Symmetric quivers and arrow reversal #

This file contains constructions related to symmetric quivers:

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].

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class Quiver.HasReverse (V : Type u_2) [Quiver V] :
Type (max u_2 v)

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

  • reverse' : {a b : V} → (a b)(b a)

    the map which sends an arrow to its reverse

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def Quiver.reverse {V : Type u_4} [Quiver V] [Quiver.HasReverse V] {a : V} {b : V} :
(a b)(b a)

Reverse the direction of an arrow.

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  • Quiver.reverse = Quiver.HasReverse.reverse'
class Quiver.HasInvolutiveReverse (V : Type u_2) [Quiver V] extends Quiver.HasReverse :
Type (max u_2 v)

A quiver HasInvolutiveReverse if reversing twice is the identity.

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@[simp]
theorem Quiver.reverse_reverse {V : Type u_2} [Quiver V] [h : Quiver.HasInvolutiveReverse V] {a : V} {b : V} (f : a b) :
@[simp]
theorem Quiver.reverse_inj {V : Type u_2} [Quiver V] [h : Quiver.HasInvolutiveReverse V] {a : V} {b : V} (f : a b) (g : a b) :
theorem Quiver.eq_reverse_iff {V : Type u_2} [Quiver V] [h : Quiver.HasInvolutiveReverse V] {a : V} {b : V} (f : a b) (g : b a) :
class Prefunctor.MapReverse {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] [Quiver.HasReverse U] [Quiver.HasReverse V] (φ : U ⥤q V) :

A prefunctor preserving reversal of arrows

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@[simp]
theorem Prefunctor.map_reverse {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] [Quiver.HasReverse U] [Quiver.HasReverse V] (φ : U ⥤q V) [Prefunctor.MapReverse φ] {u : U} {v : U} (e : u v) :
φ.map (Quiver.reverse e) = Quiver.reverse (φ.map e)
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@[simp]
@[inline, reducible]
abbrev Quiver.Hom.toPos {V : Type u_2} [Quiver V] {X : V} {Y : V} (f : X Y) :
X Y

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

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@[inline, reducible]
abbrev Quiver.Hom.toNeg {V : Type u_2} [Quiver V] {X : V} {Y : V} (f : X Y) :
Y X

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

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def Quiver.Path.reverse {V : Type u_2} [Quiver V] [Quiver.HasReverse V] {a : V} {b : V} :

Reverse the direction of a path.

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The inclusion of a quiver in its symmetrification

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  • Quiver.Symmetrify.of = { obj := id, map := fun {X Y : V} => Sum.inl }
def Quiver.Symmetrify.lift {V : Type u_2} [Quiver V] {V' : Type u_4} [Quiver V'] [Quiver.HasReverse V'] (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'

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  • One or more equations did not get rendered due to their size.
theorem Quiver.Symmetrify.lift_spec {V : Type u_2} [Quiver V] {V' : Type u_4} [Quiver V'] [Quiver.HasReverse V'] (φ : V ⥤q V') :
Quiver.Symmetrify.of ⋙q Quiver.Symmetrify.lift φ = φ
theorem Quiver.Symmetrify.lift_unique {V : Type u_2} [Quiver V] {V' : Type u_4} [Quiver V'] [Quiver.HasReverse V'] (φ : V ⥤q V') (Φ : Quiver.Symmetrify V ⥤q V') (hΦ : Quiver.Symmetrify.of ⋙q Φ = φ) (hΦinv : ∀ {X Y : Quiver.Symmetrify V} (f : X Y), Φ.map (Quiver.reverse f) = Quiver.reverse (Φ.map f)) :

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

@[simp]
theorem Prefunctor.symmetrify_map {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (φ : U ⥤q V) :
∀ {X Y : Quiver.Symmetrify U} (a : (X Y) (Y X)), (Prefunctor.symmetrify φ).map a = Sum.map φ.map φ.map a
@[simp]
theorem Prefunctor.symmetrify_obj {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (φ : U ⥤q V) :
∀ (a : U), (Prefunctor.symmetrify φ).obj a = φ.obj a
def Prefunctor.symmetrify {U : Type u_1} {V : Type u_2} [Quiver U] [Quiver V] (φ : U ⥤q V) :

A prefunctor canonically defines a prefunctor of the symmetrifications.

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theorem Quiver.Push.of_reverse {V : Type u_2} [Quiver V] {V' : Type u_4} (σ : VV') [Quiver.HasInvolutiveReverse V] (X : V) (Y : V) (f : X Y) :
instance Quiver.Push.ofMapReverse {V : Type u_2} [Quiver V] {V' : Type u_4} (σ : VV') [h : Quiver.HasInvolutiveReverse V] :
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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 SimpleGraph, since a path in one direction doesn't induce one in the other.

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