instIsLocallySurjectiveHomObjForgetTypeYonedaGluedToSheafOfIsLocallySurjectiveZariskiTopologyDescFunctorOpposite

English

If the presheaf descends along Sigma with surjectivity, then the glued-to-sheaf presheaf is locally surjective.

Русский

Если тождественная связка спадaет по Sigma и локально сюръективна, то glue-to-sheaf является локально сюръективной.

LaTeX

$$$\\text{Presheaf.IsLocallySurjective} \\bigl(\\Sigma.desc f\\bigr) \\implies \\text{Sheaf.IsLocallySurjective} (\\yonedaGluedToSheaf hf)$$$

Lean4

instance [Presheaf.IsLocallySurjective Scheme.zariskiTopology (Sigma.desc f)] :
    Sheaf.IsLocallySurjective (yonedaGluedToSheaf hf) :=
  Presheaf.isLocallySurjective_of_isLocallySurjective_fac _
    (show Sigma.desc (fun i ↦ yoneda.map (toGlued hf i)) ≫ (yonedaGluedToSheaf hf).val = Sigma.desc f by cat_disch)

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