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.
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
The dual concept is that of regular monomorphism.
The definition refers only to some parallel pair, but often there is a canonical choice: the kernel pair of the morphism in question. A morphism having a kernel pair (such as any morphism in a category with pullbacks) is a regular epimorphism if and only if it is the coequalizer of its kernel pair. See, for instance Lemma 5.6.6 in Practical Foundations; this also follows from the theory of generalized kernels). A regular epimorphism with a kernel pair, or equivalently a morphism that is the coequalizer of its kernel pair, is here called an effective epimorphism.
Although the definition doesn't state so explicitly, it is true (and easy to prove) that any regular epimorphism 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 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).
In a regular category, regular epimorphisms are preserved by pullback.
In a regular category, if in a pullback diagram
the two bottom morphisms are regular epis (hence by prop. 1 all four morphisms are), then the diagram is also a pushout.
(e.g. Johnstone, section A, prop. 1.4.3)
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.)
Regular epimorphisms are not generally closed under composition. Paul Taylor in Practical Foundations of Mathematics, 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
is not a regular epi in $Cat$. A very similar example is found at Bednarczyk et al., Example 4.4.
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 (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 smallest 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.
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:
Peter Johnstone, Section A Sketches of an Elephant
Marek A. Bednarczyk, Andrzej M. Borzyszkowski, and Wieslaw Pawlowski, Generalized congruences – epimorphisms in $\mathcal{C}a t$, Theory and Applications of Categories [electronic only] 5 (1999), 266-280. https://eudml.org/doc/120226
Paul Taylor, Practical Foundations of Mathematics, Cambridge Studies in Advanced Mathematics 59, Cambridge University Press 1999.