sheafBotEquivalence — Mathlib

English

If e: P ≅ P' is an isomorphism, the sheaf property transfers between P and P'.

Русский

Если e: P ≅ P' изоморфизм, свойство шейфа передаётся между P и P'.

LaTeX

$$$(\\text{IsSheaf } J\\ P) \\iff (\\text{IsSheaf } J\\ P').$$$

Lean4

/-- The category of sheaves on the bottom (trivial) Grothendieck topology is
equivalent to the category of presheaves.
-/
@[simps]
def sheafBotEquivalence : Sheaf (⊥ : GrothendieckTopology C) A ≌ Cᵒᵖ ⥤ A
    where
  functor := sheafToPresheaf _ _
  inverse :=
    { obj := fun P => ⟨P, Presheaf.isSheaf_bot P⟩
      map := fun f => ⟨f⟩ }
  unitIso := Iso.refl _
  counitIso := Iso.refl _

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