Epimorphism

The term epimorphism (epi - on and Greek μορφή morph of Greek ἐπί shape, form ) is used differently in mathematics. In universal algebra, an epimorphism is a homomorphism, which is surjective. In category theory epimorphism is the dual notion to monomorphism and generalizes the ( set-theoretic ) notion of surjective map.

The two terms are equivalent, at least in the following cases:

• Vector spaces, or more generally modules
• ( abelian ) groups

• 2.1 Examples

Epimorphism in the category theory

Definition

In category theory, an epimorphism is a morphism with the following property:

Y (together with f) is then called a quotient object of X.

In the arrow diagrams of homological algebra is an epimorphism as a short exact sequence

Or using a two- tip arrow with two terms as

Noted.

Special epimorphisms

An epimorphism f is called extremal if it is epimorphism and satisfies the following additional extremal:

Examples

Epimorphisms of vector spaces or modules as well as general ( abelian ) groups are exactly the surjective homomorphisms.

Epimorphisms of rings are in general not surjective, see below

In the categories Set, Grp the epimorphisms are precisely the extremal epimorphisms, namely the surjective morphisms.

In the category of topological spaces to the epimorphisms surjective continuous mappings and extremal epimorphisms are the quotient maps.

In the category Top2 of Hausdorff spaces the extremal epimorphisms are the same as in top, however, the epimorphisms are the continuous maps with dense image. This fact is often exploited in so-called " density circuits ": To show that two continuous functions dom ( a Hausdorff space ) are already common domain of definition, it suffices to show that they coincide on a dense subset D of the domain. The inclusion mapping D → dom is an epimorphism, from which the equality in the whole domain of definition follows.

In the category BanSp1 the epimorphisms are the linear continuous maps with dense image ( Banach spaces are Hausdorff ) and the extremal epimorphisms are the surjective continuous linear maps.

Epimorphism in the universal algebra

In universal algebra, an epimorphism is defined as surjective homomorphism.

Examples

Is a homomorphism, so is surjective, so an epimorphism.

There is a canonical epimorphism p for every normal subgroup N of a group G: G → G / N, the mapping g is an element of G to its coset gN.

The best-known examples of canonical epimorphisms are the pictures of an integer assigns its remaining division by a natural number m, where this radical is understood as an element of the residue class ring Z / mZ.

The parallel projection is in linear algebra is a vector space homomorphism which maps a vector space is surjective onto a subspace.

Example: not surjective Ringepimorphismus

For rings, the top two definitions are not compatible. Consider the embedding of the integers in the rational numbers:

She is not surjective and thus no epimorphism in the sense of universal algebra. However, it is an epimorphism in the category of commutative rings with identity.

Proof: We want to show that is rechtskürzbar. Be an arbitrary commutative ring with unity and let

Two ring homomorphisms. We assume that the linkages and, therefore, the limitations of and to correspond, ie

For all integers. Let and integers. We note first that because

Is a unit in. Thus, one may in

By dividing, it is therefore

Accordingly for. This follows

We have thus shown that the images are identical and, therefore proved that is rechtskürzbar. The illustration is an epimorphism in the categorical sense.

Q.e.d.