epi_iff_locallySurjective_on_compHaus

English

The epi_iff_surjective_on_stonean holds for Condensed sets and Stonean objects.

Русский

Справедливостe epi на Stonean: эпи-ограничения эквивалентны сюръективности на каждом Stonean объекте.

LaTeX

$$$\\mathrm{epi_iff_surjective_on_stonean}(f) \\iff \\forall S:\\mathrm{Stonean},\\; \\mathrm{Surjective}((f)_{S})$$$

Lean4

theorem epi_iff_locallySurjective_on_compHaus :
    Epi f ↔
      ∀ (S : CompHaus) (y : ToType (Y.val.obj ⟨S⟩)),
        (∃ (S' : CompHaus) (φ : S' ⟶ S) (_ : Function.Surjective φ) (x : ToType (X.val.obj ⟨S'⟩)),
          f.val.app ⟨S'⟩ x = Y.val.map ⟨φ⟩ y) :=
  by
  rw [← isLocallySurjective_iff_epi', coherentTopology.isLocallySurjective_iff, regularTopology.isLocallySurjective_iff]
  simp_rw [((CompHaus.effectiveEpi_tfae _).out 0 2 :)]

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