toPresieveOver_le_arrows_iff — Mathlib

English

Compatiblity of glued components with arrows in the Yoneda framework.

Русский

Совместимость взвешенных компонент с стрелками в фреймворке Йонеда.

LaTeX

$$$$ (\\yonedaGluedToSheaf hf).val.app { unop := V } (\\gamma ≫ α) = F.val.map γ^{op} ((\\yonedaGluedToSheaf hf).val.app { unop := U } α) $$$$

Lean4

theorem toPresieveOver_le_arrows_iff {X : Over S} (R : Sieve X) (𝒰 : Cover.{u} (precoverage P) X.left) [𝒰.Over S] :
    𝒰.toPresieveOver ≤ R.arrows ↔ Presieve.ofArrows 𝒰.X 𝒰.f ≤ (Sieve.overEquiv X R).arrows :=
  by
  simp_rw [← Sieve.giGenerate.gc.le_iff_le, ← Sieve.overEquiv_le_overEquiv_iff]
  rw [overEquiv_generate_toPresieveOver_eq_ofArrows]

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