Incategory theory, anepimorphism is amorphismf :X →Y that isright-cancellative in the sense that, for all objectsZ and all morphismsg₁,g₂:Y →Z,

${\displaystyle g_1\circ f=g_2\circ f⟹g_1=g_2.}$

Epimorphisms are categorical analogues ofonto or surjective functions (and in thecategory of sets the concept corresponds exactly to the surjective functions), but they may not exactly coincide in all contexts; for example, the inclusion${\displaystyle \mathbb{Z}\to \mathbb{Q}}$ is a ring epimorphism. Thedual of an epimorphism is amonomorphism (i.e. an epimorphism in acategoryC is a monomorphism in thedual categoryC^op).

Many authors inabstract algebra anduniversal algebra define anepimorphism simply as anonto orsurjectivehomomorphism. Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories. In this article, the term "epimorphism" will be used in the sense of category theory given above. For more on this, see§ Terminology below.

Every morphism in aconcrete category whose underlyingfunction issurjective is an epimorphism. In many concrete categories of interest the converse is also true. For example, in the following categories, the epimorphisms are exactly those morphisms that are surjective on the underlying sets:

• Set:sets and functions. To prove that every epimorphismf:X →Y inSet is surjective, we compose it with both thecharacteristic functiong₁:Y → {0,1} of the imagef(X) and the mapg₂:Y → {0,1} that is constant 1.
• Rel: sets withbinary relations and relation-preserving functions. Here we can use the same proof as forSet, equipping {0,1} with the full relation {0,1}×{0,1}.
• Pos:partially ordered sets andmonotone functions. Iff : (X, ≤) → (Y, ≤) is not surjective, picky₀ inY \f(X) and letg₁ :Y → {0,1} be the characteristic function of {y |y₀ ≤y} andg₂ :Y → {0,1} the characteristic function of {y |y₀ <y}. These maps are monotone if {0,1} is given the standard ordering 0 < 1.
• Grp:groups andgroup homomorphisms. The result that every epimorphism inGrp is surjective is due toOtto Schreier (he actually proved more, showing that everysubgroup is anequalizer using thefree product with one amalgamated subgroup); anelementary proof can be found in (Linderholm 1970).
• FinGrp:finite groups and group homomorphisms. Also due to Schreier; the proof given in (Linderholm 1970) establishes this case as well.
• Ab:abelian groups and group homomorphisms.
• K-Vect:vector spaces over afieldK andK-linear transformations.
• Mod-R:right modules over aringR andmodule homomorphisms. This generalizes the two previous examples; to prove that every epimorphismf:X →Y inMod-R is surjective, we compose it with both the canonicalquotient mapg₁:Y →Y/f(X) and thezero mapg₂:Y →Y/f(X).
• Top:topological spaces andcontinuous functions. To prove that every epimorphism inTop is surjective, we proceed exactly as inSet, giving {0,1} theindiscrete topology, which ensures that all considered maps are continuous.
• HComp:compactHausdorff spaces and continuous functions. Iff:X →Y is not surjective, lety ∈Y −fX. SincefX is closed, byUrysohn's Lemma there is a continuous functiong₁:Y → [0,1] such thatg₁ is 0 onfX and 1 ony. We composef with bothg₁ and the zero functiong₂:Y → [0,1].

However, there are also many concrete categories of interest where epimorphisms fail to be surjective. A few examples are:

• In thecategory of monoids,Mon, theinclusion mapN →Z is a non-surjective epimorphism. To see this, suppose thatg₁ andg₂ are two distinct maps fromZ to some monoidM. Then for somen inZ,g₁(n) ≠g₂(n), sog₁(−n) ≠g₂(−n). Eithern or −n is inN, so the restrictions ofg₁ andg₂ toN are unequal.
• In the category of algebras over commutative ringR, takeR[N] →R[Z], whereR[G] is themonoid ring of the monoidG and the morphism is induced by the inclusionN →Z as in the previous example. This follows from the observation that1 generates the algebraR[Z] (note that the unit inR[Z] is given by0 ofZ), and the inverse of the element represented byn inZ is just the element represented by −n. Thus any homomorphism fromR[Z] is uniquely determined by its value on the element represented by1 ofZ.
• In thecategory of rings,Ring, the inclusion mapZ →Q is a non-surjective epimorphism; to see this, note that anyring homomorphism onQ is determined entirely by its action onZ, similar to the previous example. A similar argument shows that the natural ring homomorphism from anycommutative ringR to any one of itslocalizations is an epimorphism.
• In thecategory of commutative rings, afinitely generated homomorphism of ringsf :R →S is an epimorphism if and only if for allprime idealsP ofR, the idealQ generated byf(P) is eitherS or is prime, and ifQ is notS, the induced mapFrac(R/P) → Frac(S/Q) is anisomorphism (EGA IV 17.2.6).
• In the category of Hausdorff spaces,Haus, the epimorphisms are precisely the continuous functions withdense images. For example, the inclusion mapQ →R, is a non-surjective epimorphism.

The above differs from the case of monomorphisms where it is more frequently true that monomorphisms are precisely those whose underlying functions areinjective.

As for examples of epimorphisms in non-concrete categories:

• If amonoid orring is considered as a category with a single object (composition of morphisms given by multiplication), then the epimorphisms are precisely the right-cancellable elements.
• If adirected graph is considered as a category (objects are the vertices, morphisms are the paths, composition of morphisms is the concatenation of paths), thenevery morphism is an epimorphism.

Everyisomorphism is an epimorphism; indeed only a right-sided inverse is needed: suppose there exists a morphismj :Y →X such thatfj = id_Y. For any morphisms${\displaystyle h_1,h_2:Y\to Z}$ where${\displaystyle h_{1}f=h_{2}f}$, you have that${\displaystyle h_1=h_{1}i{d}_Y=h_{1}fj=h_{2}fj=h_2}$. A map with such a right-sided inverse is called asplit epi. In atopos, a map that is both amonic morphism and an epimorphism is an isomorphism.

The composition of two epimorphisms is again an epimorphism. If the compositionfg of two morphisms is an epimorphism, thenf must be an epimorphism.

As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context. IfD is asubcategory ofC, then every morphism inD that is an epimorphism when considered as a morphism inC is also an epimorphism inD. However the converse need not hold; the smaller category can (and often will) have more epimorphisms.

As for most concepts in category theory, epimorphisms are preserved underequivalences of categories: given an equivalenceF :C →D, a morphismf is an epimorphism in the categoryC if and only ifF(f) is an epimorphism inD. Aduality between two categories turns epimorphisms into monomorphisms, and vice versa.

The definition of epimorphism may be reformulated to state thatf :X →Y is an epimorphism if and only if the induced maps

areinjective for every choice ofZ. This in turn is equivalent to the inducednatural transformation

being a monomorphism in thefunctor categorySet^C.

Everycoequalizer is an epimorphism, a consequence of the uniqueness requirement in the definition of coequalizers. It follows in particular that everycokernel is an epimorphism. The converse, namely that every epimorphism be a coequalizer, is not true in all categories.

In many categories it is possible to write every morphism as the composition of an epimorphism followed by a monomorphism. For instance, given a group homomorphismf :G →H, we can define the groupK = im(f) and then writef as the composition of the surjective homomorphismG →K that is defined likef, followed by the injective homomorphismK →H that sends each element to itself. Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in allabelian categories and also in all the concrete categories mentioned above in§ Examples (though not in all concrete categories).

Among other useful concepts areregular epimorphism,extremal epimorphism,immediate epimorphism,strong epimorphism, andsplit epimorphism.

• An epimorphism is said to beregular if it is acoequalizer of some pair of parallel morphisms.
• An epimorphism${\displaystyle \epsilon }$ is said to beextremal^[1] if in each representation${\displaystyle \epsilon =\mu \circ \phi }$, where${\displaystyle \mu }$ is amonomorphism, the morphism${\displaystyle \mu }$ is automatically anisomorphism.
• An epimorphism${\displaystyle \epsilon }$ is said to beimmediate if in each representation${\displaystyle \epsilon =\mu \circ {\epsilon }^{\prime }}$, where${\displaystyle \mu }$ is amonomorphism and${\displaystyle {\epsilon }^{\prime }}$ is an epimorphism, the morphism${\displaystyle \mu }$ is automatically anisomorphism.

• 

  An epimorphism${\displaystyle \epsilon :A\to B}$ is said to bestrong^[1][2] if for anymonomorphism${\displaystyle \mu :C\to D}$ and any morphisms${\displaystyle \alpha :A\to C}$ and${\displaystyle \beta :B\to D}$ such that${\displaystyle \beta \circ \epsilon =\mu \circ \alpha }$, there exists a morphism${\displaystyle \delta :B\to C}$ such that${\displaystyle \delta \circ \epsilon =\alpha }$ and${\displaystyle \mu \circ \delta =\beta }$.
• An epimorphism${\displaystyle \epsilon }$ is said to besplit if there exists a morphism${\displaystyle \mu }$ such that${\displaystyle \epsilon \circ \mu =1}$ (in this case${\displaystyle \mu }$ is called a right-sided inverse for${\displaystyle \epsilon }$).

There is also the notion ofhomological epimorphism in ring theory. A morphismf:A →B of rings is a homological epimorphism if it is an epimorphism and it induces afull and faithful functor onderived categories:D(f) : D(B) → D(A).

A morphism that is both a monomorphism and an epimorphism is called abimorphism. Every isomorphism is a bimorphism but the converse is not true in general. For example, the map from thehalf-open interval [0,1) to theunit circle S¹ (thought of as asubspace of thecomplex plane) that sendsx to exp(2πix) (seeEuler's formula) is continuous and bijective but not ahomeomorphism since the inverse map is not continuous at 1, so it is an instance of a bimorphism that is not an isomorphism in the categoryTop. Another example is the embeddingQ →R in the categoryHaus; as noted above, it is a bimorphism, but it is not bijective and therefore not an isomorphism. Similarly, in the category ofrings, the mapZ →Q is a bimorphism but not an isomorphism.

Epimorphisms are used to define abstractquotient objects in general categories: two epimorphismsf₁ :X →Y₁ andf₂ :X →Y₂ are said to beequivalent if there exists an isomorphismj :Y₁ →Y₂ withjf₁ =f₂. This is anequivalence relation, and the equivalence classes are defined to be the quotient objects ofX.

The companion termsepimorphism andmonomorphism were first introduced byBourbaki. Bourbaki usesepimorphism as shorthand for asurjective function. Early category theorists believed that epimorphisms were the correct analogue of surjections in an arbitrary category, similar to how monomorphisms are very nearly an exact analogue of injections. Unfortunately this is incorrect; strong or regular epimorphisms behave much more closely to surjections than ordinary epimorphisms.Saunders Mac Lane attempted to create a distinction betweenepimorphisms, which were maps in a concrete category whose underlying set maps were surjective, andepic morphisms, which are epimorphisms in the modern sense. However, this distinction never caught on.

It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of rings. In general, epimorphisms are their own unique concept, related to surjections but fundamentally different.

• List of category theory topics
• Monomorphism

• epimorphism at thenLab
• Strong epimorphism at thenLab

• Morphisms
• Algebraic properties of elements

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