nLab
effective epimorphism

Effective epimorphisms

Idea

An effective epimorphism is a morphism cdc\to d in a category which behaves in the way that a covering is expected to behave, in the sense that ”dd is the union of the parts of cc, 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 CC is a morphism f:cdf : c \to d that has a kernel pair c× dcc \times_d c and is the quotient object of this kernel pair, in that

c× dccfd c \times_d c \;\rightrightarrows\; c \overset{f}{\to} d

is a colimit diagram (a coequalizer).

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

Sometimes we say that such morphism ff 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

References

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

Revised on May 17, 2012 08:12:25 by Urs Schreiber (82.169.65.155)