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”.
An effective epimorphism in a category $C$ is a morphism $f \colon c \to d$ that has a kernel pair $c \times_d c$ and is the quotient object of this kernel pair, in that
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.
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.
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.
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 1992, Thm IV.7.8, Borceux 1994III, Prop. 3.4.13, 3.4.15).
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 $(\infty,1)$-topos is effective epi precisely if its 0-truncation is an epimorphism (hence an effective epimorphism) in the underlying 1-topos. This is Proposition 7.2.1.14 in Higher Topos Theory.
In the category of topological spaces, open covers provide an example of effective epimorphisms. Suppose $X = \bigcup_{i \in I} U_i$ for $U_i \subset X$ open subsets. Then the gluing/pasting lemma states that continuous maps $f : X \to Y$ naturally correspond to families of continuous maps $f_i : U_i \to Y$ that agree on the intersections, so $f_i |_{U_i \cap U_j} = f_j |_{U_i \cap U_j}$. But this is precisely the statement that the induced morphism $\coprod_{i \in I} U_i \to X$ (induced by the inclusion maps $U_i \to X$) is an effective epi! Of course, arbitrary categories may not have arbitrary coproducts, hence why this definition uses a single object.
epimorphism, regular epimorphism, effective epimorphism
Original articles:
Textbook accounts:
Exposition and examples:
Discussion in toposes:
Saunders MacLane, Ieke Moerdijk, section IV.7 of Sheaves in Geometry and Logic
Francis Borceux, Section 3.3.4 of: Handbook of Categorical Algebra. Vol. 3. Categories of Sheaves, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525872]
Discussion in homotopy type theory is in
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