IsSheafEqualizerProducts — Mathlib

English

The fork is a limit cone over the equalizer diagram associated to the presheaf and the cover.

Русский

Форк является пределом относительно диаграммы равносилки, связанной с преп presheaf и покрытием.

LaTeX

$$$\mathrm{Fork} (leftRes F U, rightRes F U)$ is a limit cone for the equalizer diagram.$$

Lean4

/-- The sheaf condition for a `F : Presheaf C X` requires that the morphism
`F.obj U ⟶ ∏ᶜ F.obj (U i)` (where `U` is some open set which is the union of the `U i`)
is the equalizer of the two morphisms
`∏ᶜ F.obj (U i) ⟶ ∏ᶜ F.obj (U i) ⊓ (U j)`.
-/
def IsSheafEqualizerProducts (F : Presheaf.{v', v, u} C X) : Prop :=
  ∀ ⦃ι : Type v'⦄ (U : ι → Opens X), Nonempty (IsLimit (SheafConditionEqualizerProducts.fork F U))

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