isSheaf_yoneda_obj — Mathlib

English

In the setting with finite colimits and preadditive structure, the sheaf-to-presheaf functor preserves finite colimits.

Русский

При наличии конечных колимит и предадитивной структуры, функция from sheaf to presheaf сохраняет конечные колимиты.

LaTeX

$$$\mathrm{PreservesFiniteColimits}(\mathrm{sheafToPresheaf}(\mathrm{extensiveTopology}(C) , A))$$$

Lean4

/-- Every Yoneda-presheaf is a sheaf for the extensive topology. -/
theorem isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (extensiveTopology C) (yoneda.obj W) :=
  by
  rw [extensiveTopology, isSheaf_coverage]
  intro X R ⟨Y, α, Z, π, hR, hi⟩
  have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
  have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
  exact isSheafFor_extensive_of_preservesFiniteProducts _ _

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