instSmallStrictLimitsOfShapeOfSmallOfLocallySmall

English

Instance: If P is an object property, then P.strictLimitsOfShape J is Small whenever the ambient category is locally small and J is small.

Русский

Пусть P — свойство объектов; тогда strictLimitsOfShape(J)小 будет малым, если исходная категория локально мала и J мала.

LaTeX

$$$[ObjectProperty.Small.{w} P] \Rightarrow [LocallySmall.{w} C] [Small.{w} J] [LocallySmall.{w} J] \Rightarrow P.strictLimitsOfShape J \text{ is Small}$$$

Lean4

instance [ObjectProperty.Small.{w} P] [LocallySmall.{w} C] [Small.{w} J] [LocallySmall.{w} J] :
    ObjectProperty.Small.{w} (P.strictLimitsOfShape J) :=
  by
  refine
    small_of_surjective (f := fun (F : { F : J ⥤ P.FullSubcategory // HasLimit (F ⋙ P.ι) }) ↦
      (⟨_,
        letI := F.2;
        ⟨F.1 ⋙ P.ι, fun j ↦ (F.1.obj j).2⟩⟩))
      ?_
  rintro ⟨_, ⟨F, hF⟩⟩
  exact ⟨⟨P.lift F hF, by assumption⟩, rfl⟩

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