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category_theory.limits.shapes.regular_mono

Definitions and basic properties of regular monomorphisms and epimorphisms. #

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A regular monomorphism is a morphism that is the equalizer of some parallel pair.

We give the constructions

We also define classes regular_mono_category and regular_epi_category for categories in which every monomorphism or epimorphism is regular, and deduce that these categories are strong_mono_categorys resp. strong_epi_categorys.

@[class]
structure category_theory.regular_mono {C : Type u₁} [category_theory.category C] {X Y : C} (f : X Y) :
Type (max u₁ v₁)

A regular monomorphism is a morphism which is the equalizer of some parallel pair.

Instances of this typeclass
Instances of other typeclasses for category_theory.regular_mono
  • category_theory.regular_mono.has_sizeof_inst
@[protected, instance]

Every regular monomorphism is a monomorphism.

@[protected, instance]

Every split monomorphism is a regular monomorphism.

Equations

If f is a regular mono, then any map k : W ⟶ Y equalizing regular_mono.left and regular_mono.right induces a morphism l : W ⟶ X such that l ≫ f = k.

Equations

The second leg of a pullback cone is a regular monomorphism if the right component is too.

See also pullback.snd_of_mono for the basic monomorphism version, and regular_of_is_pullback_fst_of_regular for the flipped version.

Equations

The first leg of a pullback cone is a regular monomorphism if the left component is too.

See also pullback.fst_of_mono for the basic monomorphism version, and regular_of_is_pullback_snd_of_regular for the flipped version.

Equations

A regular monomorphism is an isomorphism if it is an epimorphism.

@[class]
structure category_theory.regular_mono_category (C : Type u₁) [category_theory.category C] :
Type (max u₁ v₁)

A regular mono category is a category in which every monomorphism is regular.

Instances of this typeclass
Instances of other typeclasses for category_theory.regular_mono_category
  • category_theory.regular_mono_category.has_sizeof_inst

In a category in which every monomorphism is regular, we can express every monomorphism as an equalizer. This is not an instance because it would create an instance loop.

Equations
@[class]
structure category_theory.regular_epi {C : Type u₁} [category_theory.category C] {X Y : C} (f : X Y) :
Type (max u₁ v₁)

A regular epimorphism is a morphism which is the coequalizer of some parallel pair.

Instances of this typeclass
Instances of other typeclasses for category_theory.regular_epi
  • category_theory.regular_epi.has_sizeof_inst
@[protected, instance]

Every regular epimorphism is an epimorphism.

@[protected, instance]

Every split epimorphism is a regular epimorphism.

Equations

If f is a regular epi, then every morphism k : X ⟶ W coequalizing regular_epi.left and regular_epi.right induces l : Y ⟶ W such that f ≫ l = k.

Equations

The second leg of a pushout cocone is a regular epimorphism if the right component is too.

See also pushout.snd_of_epi for the basic epimorphism version, and regular_of_is_pushout_fst_of_regular for the flipped version.

Equations

The first leg of a pushout cocone is a regular epimorphism if the left component is too.

See also pushout.fst_of_epi for the basic epimorphism version, and regular_of_is_pushout_snd_of_regular for the flipped version.

Equations

A regular epimorphism is an isomorphism if it is a monomorphism.

@[class]
structure category_theory.regular_epi_category (C : Type u₁) [category_theory.category C] :
Type (max u₁ v₁)

A regular epi category is a category in which every epimorphism is regular.

Instances of this typeclass
Instances of other typeclasses for category_theory.regular_epi_category
  • category_theory.regular_epi_category.has_sizeof_inst

In a category in which every epimorphism is regular, we can express every epimorphism as a coequalizer. This is not an instance because it would create an instance loop.

Equations