English
For a nonempty chain c, membership of i in the carrier of find(c, ne, i) is equivalent to i belonging to the chainSupCarrier.
Русский
Для непустой цепи c принадлежность i носителю find(c, ne, i) эквивалентна тому, что i принадлежит chainSupCarrier.
LaTeX
$$$i\in (\mathrm{find}\; c\; ne\; i).\text{carrier} \iff i\in \operatorname{chainSupCarrier}(c)$$$
Lean4
/-- Let `F` and `G` be sheaves valued in a concrete category, whose forgetful functor reflects
isomorphisms, preserves limits and filtered colimits. Then a morphism `f : F ⟶ G` is an
isomorphism if and only if all of its stalk maps are isomorphisms.
-/
theorem isIso_iff_stalkFunctor_map_iso {F G : Sheaf C X} (f : F ⟶ G) :
IsIso f ↔ ∀ x : X, IsIso ((stalkFunctor C x).map f.1) :=
⟨fun _ x => @Functor.map_isIso _ _ _ _ _ _ (stalkFunctor C x) f.1 ((Sheaf.forget C X).map_isIso f), fun _ =>
isIso_of_stalkFunctor_map_iso f⟩
