Covering #

This file defines coverings of quivers as prefunctors that are bijective on the so-called stars and costars at each vertex of the domain.

Main definitions #

Quiver.Star u is the type of all arrows with source u;
Quiver.Costar u is the type of all arrows with target u;
Prefunctor.star φ u is the obvious function star u → star (φ.obj u);
Prefunctor.costar φ u is the obvious function costar u → costar (φ.obj u);
Prefunctor.IsCovering φ means that φ.star u and φ.costar u are bijections for all u;
Quiver.PathStar u is the type of all paths with source u;
Prefunctor.pathStar u is the obvious function PathStar u → PathStar (φ.obj u).

Main statements #

Prefunctor.IsCovering.pathStar_bijective states that if φ is a covering, then φ.pathStar u is a bijection for all u. In other words, every path in the codomain of φ lifts uniquely to its domain.

TODO #

Clean up the namespaces by renaming Prefunctor to Quiver.Prefunctor.

Tags #

Cover, covering, quiver, path, lift

source

@[reducible, inline]

abbrev Quiver.Star {U : Type u_1} [Quiver U] (u : U) :

Type (max u_1 u)

The Quiver.Star at a vertex is the collection of arrows whose source is the vertex. The type Quiver.Star u is defined to be Σ (v : U), (u ⟶ v).

Equations

Instances For

source

@[reducible, inline]

abbrev Quiver.Star.mk {U : Type u_1} [Quiver U] {u v : U} (f : u ⟶ v) :

Star u

Constructor for Quiver.Star. Defined to be Sigma.mk.

Equations

Instances For

source

@[reducible, inline]

abbrev Quiver.Costar {U : Type u_1} [Quiver U] (v : U) :

Type (max u_1 u)

The Quiver.Costar at a vertex is the collection of arrows whose target is the vertex. The type Quiver.Costar v is defined to be Σ (u : U), (u ⟶ v).

Equations

Instances For

source

@[reducible, inline]

abbrev Quiver.Costar.mk {U : Type u_1} [Quiver U] {u v : U} (f : u ⟶ v) :

Costar v

Constructor for Quiver.Costar. Defined to be Sigma.mk.

Equations

Instances For

source

def Prefunctor.star {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) :

Quiver.Star u → Quiver.Star (φ.obj u)

A prefunctor induces a map of Quiver.Star at every vertex.

Equations

Instances For

source

@[simp]

theorem Prefunctor.star_snd {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) (a✝ : Quiver.Star u) :

(φ.star u a✝).snd = φ.map a✝.snd

source

@[simp]

theorem Prefunctor.star_fst {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) (a✝ : Quiver.Star u) :

(φ.star u a✝).fst = φ.obj a✝.fst

source

def Prefunctor.costar {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) :

Quiver.Costar u → Quiver.Costar (φ.obj u)

A prefunctor induces a map of Quiver.Costar at every vertex.

Equations

Instances For

source

@[simp]

theorem Prefunctor.costar_snd {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) (a✝ : Quiver.Costar u) :

(φ.costar u a✝).snd = φ.map a✝.snd

source

@[simp]

theorem Prefunctor.costar_fst {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) (a✝ : Quiver.Costar u) :

(φ.costar u a✝).fst = φ.obj a✝.fst

source

@[simp]

theorem Prefunctor.star_apply {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) {u v : U} (e : u ⟶ v) :

φ.star u (Quiver.Star.mk e) = Quiver.Star.mk (φ.map e)

source

@[simp]

theorem Prefunctor.costar_apply {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) {u v : U} (e : u ⟶ v) :

φ.costar v (Quiver.Costar.mk e) = Quiver.Costar.mk (φ.map e)

source

theorem Prefunctor.star_comp {U : Type u_1} [Quiver U] {V : Type u_3} [Quiver V] (φ : U ⥤q V) {W : Type u_2} [Quiver W] (ψ : V ⥤q W) (u : U) :

(φ ⋙q ψ).star u = ψ.star (φ.obj u) ∘ φ.star u

source

theorem Prefunctor.costar_comp {U : Type u_1} [Quiver U] {V : Type u_3} [Quiver V] (φ : U ⥤q V) {W : Type u_2} [Quiver W] (ψ : V ⥤q W) (u : U) :

(φ ⋙q ψ).costar u = ψ.costar (φ.obj u) ∘ φ.costar u

source

structure Prefunctor.IsCovering {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) :

Prop

A prefunctor is a covering of quivers if it defines bijections on all stars and costars.

star_bijective (u : U) : Function.Bijective (φ.star u)
costar_bijective (u : U) : Function.Bijective (φ.costar u)

Instances For

source

@[simp]

theorem Prefunctor.IsCovering.map_injective {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (hφ : φ.IsCovering) {u v : U} :

Function.Injective fun (f : u ⟶ v) => φ.map f

source

theorem Prefunctor.IsCovering.comp {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) {W : Type u_3} [Quiver W] (ψ : V ⥤q W) (hφ : φ.IsCovering) (hψ : ψ.IsCovering) :

(φ ⋙q ψ).IsCovering

source

theorem Prefunctor.IsCovering.of_comp_right {U : Type u_3} [Quiver U] {V : Type u_1} [Quiver V] (φ : U ⥤q V) {W : Type u_2} [Quiver W] (ψ : V ⥤q W) (hψ : ψ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering) :

φ.IsCovering

source

theorem Prefunctor.IsCovering.of_comp_left {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) {W : Type u_3} [Quiver W] (ψ : V ⥤q W) (hφ : φ.IsCovering) (hφψ : (φ ⋙q ψ).IsCovering) (φsur : Function.Surjective φ.obj) :

ψ.IsCovering

source

def Quiver.symmetrifyStar {U : Type u_1} [Quiver U] (u : U) :

Star (Symmetrify.of.obj u) ≃ Star u ⊕ Costar u

The star of the symmetrification of a quiver at a vertex u is equivalent to the sum of the star and the costar at u in the original quiver.

Equations

Instances For

source

def Quiver.symmetrifyCostar {U : Type u_1} [Quiver U] (u : U) :

Costar (Symmetrify.of.obj u) ≃ Costar u ⊕ Star u

The costar of the symmetrification of a quiver at a vertex u is equivalent to the sum of the costar and the star at u in the original quiver.

Equations

Instances For

source

theorem Prefunctor.symmetrifyStar {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) :

φ.symmetrify.star u = ⇑(Quiver.symmetrifyStar (φ.obj u)).symm ∘ Sum.map (φ.star u) (φ.costar u) ∘ ⇑(Quiver.symmetrifyStar u)

source

theorem Prefunctor.symmetrifyCostar {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) :

φ.symmetrify.costar u = ⇑(Quiver.symmetrifyCostar (φ.obj u)).symm ∘ Sum.map (φ.costar u) (φ.star u) ∘ ⇑(Quiver.symmetrifyCostar u)

source

theorem Prefunctor.IsCovering.symmetrify {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (hφ : φ.IsCovering) :

φ.symmetrify.IsCovering

source

@[reducible, inline]

abbrev Quiver.PathStar {U : Type u_1} [Quiver U] (u : U) :

Type (max u_1 u u_1)

The path star at a vertex u is the type of all paths starting at u. The type Quiver.PathStar u is defined to be Σ v : U, Path u v.

Equations

Instances For

source

@[reducible, inline]

abbrev Quiver.PathStar.mk {U : Type u_1} [Quiver U] {u v : U} (p : Path u v) :

PathStar u

Constructor for Quiver.PathStar. Defined to be Sigma.mk.

Equations

Instances For

source

def Prefunctor.pathStar {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (u : U) :

Quiver.PathStar u → Quiver.PathStar (φ.obj u)

A prefunctor induces a map of path stars.

Equations

Instances For

source

@[simp]

theorem Prefunctor.pathStar_apply {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) {u v : U} (p : Quiver.Path u v) :

φ.pathStar u (Quiver.PathStar.mk p) = Quiver.PathStar.mk (φ.mapPath p)

source

theorem Prefunctor.pathStar_injective {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (hφ : ∀ (u : U), Function.Injective (φ.star u)) (u : U) :

Function.Injective (φ.pathStar u)

source

theorem Prefunctor.pathStar_surjective {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (hφ : ∀ (u : U), Function.Surjective (φ.star u)) (u : U) :

Function.Surjective (φ.pathStar u)

source

theorem Prefunctor.pathStar_bijective {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) (hφ : ∀ (u : U), Function.Bijective (φ.star u)) (u : U) :

Function.Bijective (φ.pathStar u)

source

theorem Prefunctor.IsCovering.pathStar_bijective {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] {φ : U ⥤q V} (hφ : φ.IsCovering) (u : U) :

Function.Bijective (φ.pathStar u)

source

def Quiver.starEquivCostar {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] (u : U) :

Star u ≃ Costar u

In a quiver with involutive inverses, the star and costar at every vertex are equivalent. This map is induced by Quiver.reverse.

Equations

Instances For

source

@[simp]

theorem Quiver.starEquivCostar_symm_apply_fst {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] (u : U) (e : Costar u) :

((starEquivCostar u).symm e).fst = e.fst

source

@[simp]

theorem Quiver.starEquivCostar_symm_apply_snd {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] (u : U) (e : Costar u) :

((starEquivCostar u).symm e).snd = reverse e.snd

source

@[simp]

theorem Quiver.starEquivCostar_apply_snd {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] (u : U) (e : Star u) :

((starEquivCostar u) e).snd = reverse e.snd

source

@[simp]

theorem Quiver.starEquivCostar_apply_fst {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] (u : U) (e : Star u) :

((starEquivCostar u) e).fst = e.fst

source

@[simp]

theorem Quiver.starEquivCostar_apply {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] {u v : U} (e : u ⟶ v) :

(starEquivCostar u) (Star.mk e) = Costar.mk (reverse e)

source

@[simp]

theorem Quiver.starEquivCostar_symm_apply {U : Type u_1} [Quiver U] [HasInvolutiveReverse U] {u v : U} (e : u ⟶ v) :

(starEquivCostar v).symm (Costar.mk e) = Star.mk (reverse e)

source

theorem Prefunctor.costar_conj_star {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) [Quiver.HasInvolutiveReverse U] [Quiver.HasInvolutiveReverse V] [φ.MapReverse] (u : U) :

φ.costar u = ⇑(Quiver.starEquivCostar (φ.obj u)) ∘ φ.star u ∘ ⇑(Quiver.starEquivCostar u).symm

source

theorem Prefunctor.bijective_costar_iff_bijective_star {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) [Quiver.HasInvolutiveReverse U] [Quiver.HasInvolutiveReverse V] [φ.MapReverse] (u : U) :

Function.Bijective (φ.costar u) ↔ Function.Bijective (φ.star u)

source

theorem Prefunctor.isCovering_of_bijective_star {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) [Quiver.HasInvolutiveReverse U] [Quiver.HasInvolutiveReverse V] [φ.MapReverse] (h : ∀ (u : U), Function.Bijective (φ.star u)) :

φ.IsCovering

source

theorem Prefunctor.isCovering_of_bijective_costar {U : Type u_1} [Quiver U] {V : Type u_2} [Quiver V] (φ : U ⥤q V) [Quiver.HasInvolutiveReverse U] [Quiver.HasInvolutiveReverse V] [φ.MapReverse] (h : ∀ (u : U), Function.Bijective (φ.costar u)) :

φ.IsCovering

Documentation

Mathlib.Combinatorics.Quiver.Covering

Covering #

Main definitions #

Main statements #

TODO #

Tags #