Pushing a quiver structure along a map

Given a map σ : V → W→ W and a Quiver instance on V, this files defines a quiver instance on W by associating to each arrow v ⟶ v'⟶ v' in V an arrow σ v ⟶ σ v'⟶ σ v' in W.

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def Quiver.Push {V : Type u_1} {W : Type u_2} : (V → W) → Type u_2

The Quiver instance obtained by pushing arrows of V along the map σ : V → W→ W

Equations

• Quiver.Push x = W

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instance Quiver.instNonemptyPush {V : Type u_1} {W : Type u_2} (σ : V → W) [h : Nonempty W] : Nonempty (Quiver.Push σ)

Equations

• @Quiver.instNonemptyPush = @Quiver.instNonemptyPush.proof_1

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inductive Quiver.PushQuiver {V : Type u} [inst : Quiver V] {W : Type u₂} (σ : V → W) : W → W → Type (maxuu₂v)

• arrow: {V : Type u} → [inst : Quiver V] → {W : Type u₂} → {σ : V → W} → {X Y : V} → (X ⟶ Y) → Quiver.PushQuiver σ (σ X) (σ Y)

The quiver structure obtained by pushing arrows of V along the map σ : V → W→ W

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instance Quiver.instQuiverPush {V : Type u_1} [inst : Quiver V] {W : Type u_3} (σ : V → W) : Quiver (Quiver.Push σ)

Equations

• Quiver.instQuiverPush σ = { Hom := Quiver.PushQuiver σ }

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def Quiver.Push.of {V : Type u_1} [inst : Quiver V] {W : Type u_3} (σ : V → W) : V ⥤q Quiver.Push σ

The prefunctor induced by pushing arrows via σ

Equations

• Quiver.Push.of σ = { obj := σ, map := fun {X Y} f => Quiver.PushQuiver.arrow f }

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@[simp]

theorem Quiver.Push.of_obj {V : Type u_1} [inst : Quiver V] {W : Type u_2} (σ : V → W) : (Quiver.Push.of σ).obj = σ

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noncomputable def Quiver.Push.lift {V : Type u_1} [inst : Quiver V] {W : Type u_3} (σ : V → W) {W' : Type u_4} [inst : Quiver W'] (φ : V ⥤q W') (τ : W → W') (h : ∀ (x : V), Prefunctor.obj φ x = τ (σ x)) : Quiver.Push σ ⥤q W'

Given a function τ : W → W'→ W' and a prefunctor φ : V ⥤q W'⥤q W', one can extend τ to be a prefunctor W ⥤q W'⥤q W' if τ and σ factorize φ at the level of objects, where W is given the pushforward quiver structure Push σ.

Equations

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

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theorem Quiver.Push.lift_obj {V : Type u_3} [inst : Quiver V] {W : Type u_1} (σ : V → W) {W' : Type u_2} [inst : Quiver W'] (φ : V ⥤q W') (τ : W → W') (h : ∀ (x : V), Prefunctor.obj φ x = τ (σ x)) : (Quiver.Push.lift σ φ τ h).obj = τ

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theorem Quiver.Push.lift_comp {V : Type u_1} [inst : Quiver V] {W : Type u_5} (σ : V → W) {W' : Type u_3} [inst : Quiver W'] (φ : V ⥤q W') (τ : W → W') (h : ∀ (x : V), Prefunctor.obj φ x = τ (σ x)) : Quiver.Push.of σ ⋙q Quiver.Push.lift σ φ τ h = φ

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theorem Quiver.Push.lift_unique {V : Type u_1} [inst : Quiver V] {W : Type u_3} (σ : V → W) {W' : Type u_5} [inst : Quiver W'] (φ : V ⥤q W') (τ : W → W') (h : ∀ (x : V), Prefunctor.obj φ x = τ (σ x)) (Φ : Quiver.Push σ ⥤q W') (Φ₀ : Φ.obj = τ) (Φcomp : Quiver.Push.of σ ⋙q Φ = φ) : Φ = Quiver.Push.lift σ φ τ h