IsSheafPreservesLimitPairwiseIntersections

English

There is a natural isomorphism between whiskered cones of OpensLeCover and Sieve-generated cones after applying generate and related equivalences.

Русский

Существует естественный изоморфизм между взбитыми конусами OpensLeCover и коконами, порождаемыми сидом, после применения generate и связанных эквивалентностей.

LaTeX

$$$\text{whiskerIsoMapGenerateCocone}: \text{F.mapCone opensLeCover} \cong \text{F.mapCone sieveGenerated}$$$

Lean4

/-- An alternative formulation of the sheaf condition
(which we prove equivalent to the usual one below as
`isSheaf_iff_isSheafPreservesLimitPairwiseIntersections`).

A presheaf is a sheaf if `F` preserves the limit of `Pairwise.diagram U`.
(Recall `Pairwise.diagram U` is the diagram consisting of the pairwise intersections
`U i ⊓ U j` mapping into the open sets `U i`. This diagram has limit `iSup U`.)
-/
def IsSheafPreservesLimitPairwiseIntersections (F : Presheaf C X) : Prop :=
  ∀ ⦃ι : Type w⦄ (U : ι → Opens X), PreservesLimit (Pairwise.diagram U).op F

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