isColimitEquivIsInitialOfIsEmpty — Mathlib

English

If X is initial and f: Y → X, then f is epi.

Русский

Если X начальный и f: Y → X, то f — эпиморфизм.

LaTeX

$$$(t:\\IsInitial X)\\ (f: Y \\to X) \\Rightarrow \\mathrm{Epi}(f)$$$

Lean4

/-- If `F` is an empty diagram,
then a cocone over `F` is colimiting iff the cocone point is initial. -/
noncomputable def isColimitEquivIsInitialOfIsEmpty {J : Type*} [Category J] [IsEmpty J] {F : J ⥤ C} (c : Cocone F) :
    IsColimit c ≃ IsInitial c.pt :=
  (IsColimit.whiskerEquivalenceEquiv (equivalenceOfIsEmpty (Discrete PEmpty.{1}) _)).trans
    (isColimitEmptyCoconeEquiv _ _ _ (.refl _))

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