nLab effective epimorphism

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”.

Definition

Definition

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

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

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.

Remark

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.

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

• 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.

Related concepts

• epimorphism, regular epimorphism, effective epimorphism

• effective epimorphism in an (∞,1)-category

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:

• Francis Borceux, Def. 2.5.3 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

Exposition and examples:

• Qiaochu Yuan, Regular and effective monomorphisms and epimorphisms, 2012

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

• Univalent Foundations Project, section 7.6 of Homotopy Type Theory – Univalent Foundations of Mathematics