nLab effective epimorphism

Effective epimorphisms

Effective epimorphisms


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”.



An effective epimorphism in a category CC is a morphism f:cdf \colon 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.


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 (see there) and a strict epimorphism. For morphisms without kernel pairs, the notion of effective epimorphism is of questionable usefulness.


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.



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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