An effective 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”.

A morphism with a kernel pair (such as any morphism in a category with pullbacks) is an effective epimorphism if and only if it is a regular epimorphism and a strict epimorphism. For morphisms without kernel pairs, the notion of effective epimorphism is of questionable usefulness.

Every effective epimorphism is, of course, a regular epimorphism and hence a strict epimorphism. Conversely, a strict epimorphism which has a kernel pair is necessarily an effective epimorphism. (This is a special case of the theory of generalized kernels.) For this reason, some writers use “effective epimorphism” in general to mean what is here called a strict epimorphism.

Examples

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

In an (∞,1)-topos the bare notion of epimorphism disappears, and effective epimorphism in an (∞,1)-category becomes the default notion of epiness. A morphism in an $(\infty,1)$-topos is effective epi precisely if its 0-truncation is an epimorphism (hence an effective epimorphism) in the underlying 1-topos.