nLab
effective epimorphism

Idea

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.

Definition

An effective epimorphism in a category $C$ is a morphism $f:c\to d$ that has a kernel pair $c{\times }_{d}c$ and is the quotient object of this kernel pair, in that

$$c{\times }_{d}c\rightrightarrows c\stackrel{f}{\to }d$$c \times_d c \;\rightrightarrows\; c \overset{f}{\to} d

is a colimit diagram (a coequalizer).

In other words, this says that $f:c\to d$ is effective if $d$ is the coimage of $f$.

Sometimes we say that such morphism $f$ is an effective quotient.

The dual concept is that of effective monomorphism.

Properties

Relation to other notions of epimorphism

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.

• More generally, in any pretopos, hence in particular in every topos, every epimorphism is an effective epimorphism. See, for instance, (MacLane-Moerdijk, theorem IV.7.8).

• 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 $(\mathrm{\infty },1)$-topos is effective epi precisely if its 0-truncation is an epimorphism (hence an effective epimorphism) in the underlying 1-topos.

Related concepts

• epimorphism, regular epimorphism, effective epimorphism

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

References

In toposes effective epimorphisms are considered in section IV.7 of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic