mathlib documentation

category_theory.limits.shapes.regular_mono

Definitions and basic properties of regular monomorphisms and epimorphisms.

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

We give the constructions

@[instance]

Every regular monomorphism is a monomorphism.

@[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
@[instance]

Every regular epimorphism is an epimorphism.

@[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