finite_limit_eq_finset_univ_inf — Mathlib

English

For a functor F from a finite J into α, the limit equals the infimum of the image: lim F = inf_j F(j).

Русский

Для функтору F : J → α с конечной диаграммой J предел равен инфимума изображения: lim F = inf_j F(j).

LaTeX

$$$\lim F = \bigwedge_{j\in J} F(j)$$$

Lean4

/-- The limit of a functor from a finite diagram into a `SemilatticeInf` with `OrderTop` is the
infimum of the objects in the image.
-/
theorem finite_limit_eq_finset_univ_inf [SemilatticeInf α] [OrderTop α] (F : J ⥤ α) : limit F = Finset.univ.inf F.obj :=
  (IsLimit.conePointUniqueUpToIso (limit.isLimit F) (finiteLimitCone F).isLimit).to_eq

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