English
Ind-objects are precisely those presheaves A for which CostructuredArrow yoneda A is filtered and finally small.
Русский
Ind-объекты совпадают с прешепафетами A, для которых CostructuredArrow yoneda A фильтрован и finally small.
LaTeX
$$$\\mathrm{IsIndObject}(A) \\iff \\big( \\mathrm{IsFiltered}(\\mathrm{CostructuredArrow}(\\mathrm{yoneda},A)) \\land \\mathrm{FinallySmall}(\\mathrm{CostructuredArrow}(\\mathrm{yoneda},A)) \\big)$$$
Lean4
/-- The recognition theorem for ind-objects: `A : Cᵒᵖ ⥤ Type v` is an ind-object if and only if
`CostructuredArrow yoneda A` is filtered and finally `v`-small.
Theorem 6.1.5 of [Kashiwara2006] -/
theorem isIndObject_iff (A : Cᵒᵖ ⥤ Type v) :
IsIndObject A ↔ (IsFiltered (CostructuredArrow yoneda A) ∧ FinallySmall.{v} (CostructuredArrow yoneda A)) :=
⟨fun h => ⟨h.isFiltered, h.finallySmall⟩, fun ⟨_, _⟩ => isIndObject_of_isFiltered_of_finallySmall A⟩
