sheafifyMap_sheafifyLift — Mathlib

English

A dual extensionality statement: equality after postcomposition with toSheafify implies equality of morphisms from the sheafified domain.

Русский

Двойственная(ext) формула: равенство после посткомпозиции с toSheafify влечет равенство морфизмов из сферы шейфования.

LaTeX

$$toSheafify J P ∘ η = toSheafify J P ∘ γ → η = γ$$

Lean4

@[reassoc (attr := simp)]
theorem sheafifyMap_sheafifyLift {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (hR : Presheaf.IsSheaf J R) :
    sheafifyMap J η ≫ sheafifyLift J γ hR = sheafifyLift J (η ≫ γ) hR :=
  by
  apply sheafifyLift_unique
  rw [← Category.assoc, ← toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]

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