nLab regular epimorphism

Idea

A regular epimorphism is a morphism $c\to d$ in a category which behaves in the way that a covering is expected to behave, in the sense that “$d$ is the union of the parts of $c$, identified with each other in some specified way”.

In a category with pullbacks, regular epimorphisms are the same as effective epimorphisms and strict epimorphisms. Every regular epimorphism is an epimorphism, and every split epimorphism is regular, but the converses frequently fail.

Definition

Definition

A regular epimorphism is a morphism $f \colon c \to d$ (in a given category) that is the coequalizer of some parallel pair of morphisms, i.e. if there exists some colimit diagram of the form

$$a \;\rightrightarrows\; c \xrightarrow{\;f\;} d.$$

Remark

The dual concept to Def. is that of regular monomorphism.

Properties

Epimorphy

Although Def. does not state it explicitly, it is true (and easy to prove) that:

Proposition

Every regular epimorphism (Def. ) is an epimorphism.

In fact, every regular epimorphism is a strong epimorphism, hence an extremal epimorphism. In particular, this implies that a regular epimorphism which is also a monomorphism must in fact be an isomorphism.

Frequently (such as in a regular category), every strong or extremal epimorphism is regular. Moreover, in a regular category, every regular epimorphism is pullback-stable, and therefore a descent morphism. If the category is moreover exact, or has stable reflexive coequalizers, then every regular epimorphism is an effective descent morphism.

On the other hand, every split epimorphism is regular, but the converse holds only rarely (it is an internal form of the axiom of choice).

Effective epimorphisms

Def. refers only to some parallel pair, but often there is a canonical choice: the kernel pair of the morphism in question:

Proposition

A morphism having a kernel pair (such as any morphism in a category with pullbacks) is a regular epimorphism (Def. ) if and only if it is the coequalizer of its kernel pair.

E.g. Borceux 1994 I, Prop. 2.5.7, Taylor 1999, Lemma 5.6.6. This also follows from the theory of generalized kernels.

Definition

A regular epimorphism that satisfies the following two conditions (which are equivalent, by Prop. )

1. it has a kernel pair,

2. it is the coequalizer of its kernel pair

is called an effective epimorphism.

Pullbacks and pushouts

Proposition

In a regular category, regular epimorphisms are preserved by pullback.

This is by definition of “regular category”.

Proposition

(in regular category pullback preserves effective epimorphisms)
In a regular category, effective epimorphisms (Def. ) are preserved by pullback:

Given a commuting diagram of the form

where

• the right vertical column is both a kernel pair and a coequalizer, hence the kernel pair of a regular epimorphism,

• all parallel squares are pullbacks

then also the left vertical column is both a kernel pair and a coequalizer, hence the kernel pair of a regular epimorphism.

(Borceux 1994 II, Prop. 2.3.3)

Proof

The left column is again a kernel pair since limits commute with limits and the left bottom vertical morphism is again a regular epimorphism, by Prop. .

Therefore the left vertical column is also a coequalizer by Prop. and hence an effective epimorphism by Def. .

Proposition

In a regular category, if in a pullback diagram

$$\array{ & \swarrow & \searrow \\ & \searrow & \swarrow }$$

the two bottom morphisms are regular epis (hence by prop. all four morphisms are), then the diagram is also a pushout.

(e.g. Johnstone, section A, prop. 1.4.3)

Proposition

In a regular category $C$, regular epimorphisms and strong epimorphisms (= extremal epimorphisms since $C$ admits pullbacks) coincide; hence regular epimorphisms are closed under composition.

(See Johnstone, A.1.3.4.)

Remark

Regular epimorphisms are not generally closed under composition. Taylor 1999 p. 289 gives the following example in Cat. Let $\mathbb{2}$ be the interval category; then the coequalizer of the two object inclusions $1 \rightrightarrows \mathbb{2}$ is a regular epi $\mathbb{2} \to B\mathbb{N}$ (the codomain is the additive monoid $\mathbb{N}$ made into a 1-object category; the epi sends the non-identity arrow of $\mathbb{2}$ to $1 \in \mathbb{N}$). There is an evident regular epi $e: \mathbb{N} \to \mathbb{Z}/3$ of monoids. But the composite

$$\mathbb{2} \to B\mathbb{N} \stackrel{B e}{\to} B\mathbb{Z}/3$$

is not a regular epi in $Cat$. A very similar example is found at Bednarczyk et al., Example 4.4.

Examples

• In the category of sets, every epimorphism is regular. Thus, it can be hard to know, when generalising concepts from $\Set$ to other categories, what kind of epimorphism to use. Frequently, regular epimorphisms are a good choice. In particular, one may define a projective object (and hence the axiom of choice) using regular epimorphisms.

• More generally, in every Grothendieck topos every epimorphism is regular (e.g. Borceux 1994III, Prop. 3.4.13 on p. 218 and in fact effective, see there).

• In the category of groups, every epimorphism is regular. A number of proofs can be found in the literature; one proof that avoids case analysis is given here.

• In the category of monoids, the inclusion $\mathbb{N}\hookrightarrow\mathbb{Z}$ is an epimorphism, even though it is far from a surjection. But in this or any other algebraic category (a category of models of an algebraic theory), the morphisms whose underlying function is surjective are precisely the regular epimorphisms. Thus $\mathbb{N}\hookrightarrow\mathbb{Z}$ is not a regular epimorphism.

• Similarly, in the category of rings, not every epimorphism is regular, as shown by the example $\mathbb{Z} \hookrightarrow \mathbb{Q}$.

• In Diff, the category of smooth (paracompact) manifolds, regular epimorphisms are not as useful as in other settings. As any split epimorphism is regular, and split epimorphisms are badly behaved in $\Diff$ (for example, pullbacks of split epis do not necessarily exist), the usual procedure is to consider the largest class of arrows inside regular epis of which all pullbacks exist, namely the surjective submersions. In the setting of differentiable stacks and Lie groupoids it is surjective submersions that play the role of regular epimorphisms.

Related pages

• The page epimorphism has a list of many types of epimorphism and their relationships.

• The dual concept is regular monomorphism. In particular, in the presence of pullbacks, effective epimorphisms and strict epimorphisms are the same as regular ones.

• In higher category theory:

  • regular epimorphism in an (∞,1)-category

References

Textbook accounts:

• Jiří Adámek, Horst Herrlich, George Strecker, Def. 7.71 in: Abstract and Concrete Categories – The Joy of Cats John Wiley and Sons, New York (1990) reprinted as: Reprints in Theory and Applications of Categories 17 (2006) 1-507 (tac:tr17, book webpage)

• Francis Borceux, Def. 4.3.1 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

• Francis Borceux, Chapter 2 of: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

• Francis Borceux, Section 3.3.4 of: Handbook of Categorical Algebra. Vol. 3. Categories of Sheaves, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525872]

Exposition and examples:

• Qiaochu Yuan, Regular and effective monomorphisms and epimorphisms, 2012

See also:

• Peter Johnstone, Section A of: Sketches of an Elephant

• Marek A. Bednarczyk, Andrzej M. Borzyszkowski, Wieslaw Pawlowski, Generalized congruences – epimorphisms in $\mathcal{C}a t$, Theory and Applications of Categories 5 11 (1999) 266-280 [tac:5-11, dml:120226]

• Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999 (webpage)