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

@[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

• Quiver.Star u = ((v : U) × (u ⟶ v))

@[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

• Quiver.Star.mk f = ⟨v, f⟩

@[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

• Quiver.Costar v = ((u : U) × (u ⟶ v))

@[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

• Quiver.Costar.mk f = ⟨u, f⟩

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

• φ.star u F = Quiver.Star.mk (φ.map F.snd)

@[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

@[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

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

• φ.costar u F = Quiver.Costar.mk (φ.map F.snd)

@[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

@[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

@[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)

@[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)

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

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

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)

@[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

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

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

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

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

• Quiver.symmetrifyStar u = Equiv.sigmaSumDistrib (Quiver.Hom (Quiver.Symmetrify.of.obj u)) fun (v : Quiver.Symmetrify U) => v ⟶ Quiver.Symmetrify.of.obj u

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

• Quiver.symmetrifyCostar u = Equiv.sigmaSumDistrib (fun (u_1 : Quiver.Symmetrify U) => u_1 ⟶ Quiver.Symmetrify.of.obj u) (Quiver.Hom (Quiver.Symmetrify.of.obj u))

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)

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)

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

@[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

• Quiver.PathStar u = ((v : U) × Quiver.Path u v)

@[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

• Quiver.PathStar.mk p = ⟨v, p⟩

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

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

@[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)

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)

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)

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)

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)

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

• One or more equations did not get rendered due to their size.

@[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

@[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

@[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

@[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

@[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)

@[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)

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

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)

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

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