nLab
epimorphism

Definition

An epimorphism in a category $C$ is a morphism $f:X\to Y$ such that every contravariant hom-functor $\mathrm{Hom}(-,Z)$ sends it to an injection

$$\forall Z\in C:\mathrm{Hom}(Y,Z)\stackrel{f^{*}}{\hookrightarrow }\mathrm{Hom}(X,Z).$$\forall Z \in C : \; Hom(Y,Z) \stackrel{f^*}{\hookrightarrow} Hom(X,Z) \,.

In more elementary terms, $f:X\to Y$ is an epimorphism if, given any $g,h:Y\to Z$, $g=h$ if $f;g=f;h$:

$$\begin{array}{ccc}& & Y\\ & {\nearrow }_f& & {\searrow }^g\\ X& & & & Z\\ & {\searrow }^f& & {\nearrow }_h\\ & & Y\\ \end{array}\Rightarrow \begin{array}{ccccc}X& {\to }^f& Y& {\rightrightarrows }_h^g& Z\\ \end{array}$$\array{ & & Y \\ & \nearrow_{f} & & \searrow^{g} \\ X & & & & Z \\ & \searrow^{f} & & \nearrow_{h} \\ & & Y \\ } \quad\quad \Rightarrow \quad\quad \array{ X & \rightarrow^{f} & Y & \rightrightarrows^g_h & Z \\ }

Can anybody make this look nicer? Preferably with curved arrows in the right-hand diagram? And maybe even with little vertical equals signs to show how the diagrams commute? I can do it in XYpic, but that's not supported by iTeX.

An epimorphism in $C^{\mathrm{op}}$ is a monomorphism in $C$.

The epimorphisms in Set are the surjective functions; thus epimorphisms can be thought of as a categorical notion of surjection. However, this is frequently not quite right: in categories of sets with extra structure, epimorphisms need not be surjective (unlike the case for monomorphisms, which are usually injective). Often, though, the surjections correspond to a stronger notion of epimorphism.

Variations

There are a sequence of variations on the concept of epimorphism, from strongest to weakest:

split epimorphism $\Rightarrow$ regular epimorphism $\Rightarrow$ strict epimorphism$\Rightarrow$ strong epimorphism$\Rightarrow$ extremal epimorphism$\Rightarrow$ epimorphism.

In the category of sets, every epimorphism is regular (and even split if you believe the axiom of choice), so it can be hard to know, when generalising concepts from $Set$ to other categories, what kind of epimorphism to use.

In general, the two serious distinctions come

• Between split epimorphisms and regular ones: in very few categories are all regular epimorphisms split. Splitting of even regular epimorphisms is a form of the axiom of choice, which may be valid in Set (if you believe it) but very often fails internally.

• Between extremal epimorphisms and “plain” epimorphisms: in many categories, the plain epimorphisms are oddly behaved, but the extremal ones are what we would expect. For instance, the inclusion $\mathbb{Z}\hookrightarrow \mathbb{Q}$ is an epimorphism of rings, but the extremal epimorphisms of rings are just the surjective ring homomorphisms.

The remaining distinctions frequently collapse. For instance:

• In a category with pullbacks, any strict epimorphism is regular, and any extremal epimorphism is strong.

• In a regular category, every extremal epimorphism is regular, so the sequence reduces to three: split, regular = strict = strong = extremal, plain. In algebraic categories (categories of algebra for a Lawvere theory), which are regular, the regular/strict/strong/extremal epimorphisms are the morphisms whose underlying function is surjective.

• In a pretopos (hence also in a topos), every epimorphism is regular, so the only distinction remaining is split versus non-split.

Moreover, even in non-regular categories, there seems to be a strong tendency for strong/extremal epimorphisms to coincide with regular/strict ones. For example, this is the case in Top. However, the distinction is real; for instance, in the category generated by the following graph:

$$\begin{array}{ccccc}& & & & C\\ & & & {}^f\nearrow \\ A& \underset{h}{\overset{k}{\rightrightarrows }}& B\\ & & & {}_g\searrow \\ & & & & D\end{array}$$\array{ &&&& C\\ &&& ^f\nearrow\\ A& \underoverset{h}{k}{\rightrightarrows} & B \\ &&& _g\searrow\\ &&&& D}

subject to the equations $fh=fk$ and $gh=gk$, both $f$ and $g$ are strong, but not strict, epimorphisms.