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category_theory.limits.shapes.normal_mono.basic

Definitions and basic properties of normal monomorphisms and epimorphisms. #

A normal monomorphism is a morphism that is the kernel of some other morphism.

We give the construction normal_mono → regular_mono (category_theory.normal_mono.regular_mono) as well as the dual construction for normal epimorphisms. We show equivalences reflect normal monomorphisms (category_theory.equivalence_reflects_normal_mono), and that the pullback of a normal monomorphism is normal (category_theory.normal_of_is_pullback_snd_of_normal).

We also define classes normal_mono_category and normal_epi_category for classes in which every monomorphism or epimorphism is normal, and deduce that these categories are regular_mono_categorys resp. regular_epi_categorys.

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

A normal monomorphism is a morphism which is the kernel of some morphism.

Instances for category_theory.normal_mono
  • category_theory.normal_mono.has_sizeof_inst

If F is an equivalence and F.map f is a normal mono, then f is a normal mono.

Equations

If f is a normal mono, then any map k : W ⟶ Y such that k ≫ normal_mono.g = 0 induces a morphism l : W ⟶ X such that l ≫ f = k.

Equations

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

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

Equations

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

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

Equations
@[class]

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

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

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

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

A normal epimorphism is a morphism which is the cokernel of some morphism.

Instances for category_theory.normal_epi
  • category_theory.normal_epi.has_sizeof_inst

If F is an equivalence and F.map f is a normal epi, then f is a normal epi.

Equations

If f is a normal epi, then every morphism k : X ⟶ W satisfying normal_epi.g ≫ k = 0 induces l : Y ⟶ W such that f ≫ l = k.

Equations

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

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

Equations

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

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

Equations
@[class]

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

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

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

Equations