isSeparator_sigma — Mathlib

English

There is a bijection between IsSeparator of the sigma object and IsSeparating of the range of the family f: β → C.

Русский

Существуют равномерные соответствия между IsSeparator у SigmaObj и IsSeparating диапазона family f: β → C.

LaTeX

$$$\\mathrm{IsSeparator}(\\Sigma f) \\iff \\mathrm{IsSeparating}(\\{\\,f(i) : i \\in \\beta\\,\\})$$$

Lean4

theorem isSeparator_sigma {β : Type w} (f : β → C) [HasCoproduct f] : IsSeparator (∐ f) ↔ IsSeparating (Set.range f) :=
  by
  refine ⟨fun h X Y u v huv => ?_, fun h => isSeparator_of_isColimit_cofan h (colimit.isColimit _)⟩
  refine h.def _ _ fun g => colimit.hom_ext fun b => ?_
  simpa using huv (f b.as) (by simp) (colimit.ι (Discrete.functor f) _ ≫ g)

Похожие утверждения