skyscraperPresheaf_isSheaf — Mathlib

English

The skyscraper presheaf is a sheaf.

Русский

Скскрайперный препешаф является шейфом.

LaTeX

$$$skyscraperPresheaf\ p_0\ A\;\text{IsSheaf}$$$

Lean4

theorem skyscraperPresheaf_isSheaf : (skyscraperPresheaf p₀ A).IsSheaf := by
  classical
    exact
    (Presheaf.isSheaf_iso_iff (eqToIso <| skyscraperPresheaf_eq_pushforward p₀ A)).mpr <|
      (Sheaf.pushforward_sheaf_of_sheaf _
        (Presheaf.isSheaf_on_punit_of_isTerminal _
          (by
            dsimp [skyscraperPresheaf]
            rw [if_neg]
            · exact terminalIsTerminal
            · #adaptation_note /-- 2024-03-24
                          Previously the universe annotation was not needed here. -/

              exact Set.notMem_empty PUnit.unit.{u + 1})))

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