English
For a pretopology K, the sheaf condition on the corresponding Grothendieck topology is equivalent to having IsSheafFor every basis presieve R ∈ K X.
Русский
Для претопологии K условие Шейфа эквивалентно тому, что IsSheafFor выполняется для каждого базисного presieve R ∈ K X.
LaTeX
$$[HasPullbacks C] (K : Pretopology C) : IsSheaf K.toGrothendieck P ↔ ∀ {X} (R : Presieve X), R ∈ K X → IsSheafFor P R$$
Lean4
theorem mk' {F : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A} (adj : F ⊣ sheafToPresheaf J A) [PreservesFiniteLimits F] : HasSheafify J A
where
isRightAdjoint := ⟨F, ⟨adj⟩⟩
isLeftExact :=
⟨by
have : (sheafToPresheaf J A).IsRightAdjoint := ⟨_, ⟨adj⟩⟩
exact fun _ _ _ ↦
preservesLimitsOfShape_of_natIso (adj.leftAdjointUniq (Adjunction.ofIsRightAdjoint (sheafToPresheaf J A)))⟩
