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})))