epi_iff_surjective_of_preservesPushout

English

Under preservation of colimits of shape WalkingSpan by forget C, epimorphism f is equivalent to the underlying map being surjective.

Русский

При сохранении колимитов формы WalkingSpan забыванием f эквивалентно эпиморфизму f и сюръективной базовой функции.

LaTeX

$$$\\operatorname{Epi}(f) \\iff \\operatorname{Surjective}(((\\mathrm{forget}\\,C).map f))$$$

Lean4

theorem epi_iff_surjective_of_preservesPushout {X Y : C} (f : X ⟶ Y) [PreservesColimitsOfShape WalkingSpan (forget C)] :
    Epi f ↔ Function.Surjective f :=
  ((forget C).epi_map_iff_epi _).symm.trans (epi_iff_surjective _)

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