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Mathlib.Topology.Category.TopCat.EffectiveEpi

Effective epimorphisms in TopCat #

This file proves the result TopCat.effectiveEpi_iff_isQuotientMap: The effective epimorphisms in TopCat are precisely the quotient maps.

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noncomputable def TopCat.effectiveEpiStructOfQuotientMap {B X : TopCat} (π : X ⟶ B) (hπ : Topology.IsQuotientMap ⇑(CategoryTheory.ConcreteCategory.hom π)) :
CategoryTheory.EffectiveEpiStruct π

Implementation: If π is a morphism in TopCat which is a quotient map, then it is an effective epimorphism. The theorem TopCat.effectiveEpi_iff_isQuotientMap should be used instead of this definition.

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Instances For
    source
    theorem TopCat.effectiveEpi_iff_isQuotientMap {B X : TopCat} (π : X ⟶ B) :
    CategoryTheory.EffectiveEpi π ↔ Topology.IsQuotientMap ⇑(CategoryTheory.ConcreteCategory.hom π)

    The effective epimorphisms in TopCat are precisely the quotient maps.

    source
    @[deprecated TopCat.effectiveEpi_iff_isQuotientMap (since := "2024-10-22")]
    theorem TopCat.effectiveEpi_iff_quotientMap {B X : TopCat} (π : X ⟶ B) :
    CategoryTheory.EffectiveEpi π ↔ Topology.IsQuotientMap ⇑(CategoryTheory.ConcreteCategory.hom π)

    Alias of TopCat.effectiveEpi_iff_isQuotientMap.

    The effective epimorphisms in TopCat are precisely the quotient maps.