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”.

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.


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.


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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