# nLab effective epimorphism

Effective epimorphisms

category theory

# Effective epimorphisms

## 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

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

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.

## References

Original articles:

• Alexander Grothendieck, p. 101 (4 of 21) in: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

Textbook accounts:

Exposition and examples:

Discussion in toposes:

Discussion in homotopy type theory is in

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