Epimorphism

Epimorphism (from the Greek ἐπί epi "on" and μορφή morphē "shape, form") is a term from the mathematical areas of algebra and category theory . In universal algebra , it denotes a homomorphism that is surjective . In category theory, epimorphism is the dual term for monomorphism and generalizes the (set-theoretical) term of surjective mapping .

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

• Vector spaces or more general modules
• ( Abelian ) groups

Epimorphism in Category Theory

definition

In category theory , an epimorphism is a morphism with the following property: ${\displaystyle f\colon X\to Y}$

If there are any morphisms with , then is always . (One also says: can be " shortened to the right ".) ${\displaystyle g,h\colon Y\to Z}$ ${\displaystyle g\circ f=h\circ f}$ ${\displaystyle g=h}$ ${\displaystyle f}$

${\displaystyle Y}$(together with ) is then called a quotient object of . ${\displaystyle f}$ ${\displaystyle X}$

In the arrow diagrams of homological algebra , an epimorphism is used as a short exact sequence ${\displaystyle f}$

${\displaystyle X\;{\overset {f}{\longrightarrow }}\;Y\longrightarrow 0}$

or using a double-headed arrow with two terms as

${\displaystyle X\;{\overset {f}{\twoheadrightarrow }}\;Y}$

written down.

Special epimorphisms

→ Main article : Extreme monomorphism and epimorphism

An epimorphism is called extremal if it is an epimorphism and also fulfills the following extremal property: ${\displaystyle f}$

If , where is a monomorphism, then there must be an isomorphism . ${\displaystyle f=m\circ g}$ ${\displaystyle m}$ ${\displaystyle m}$

Examples

Epimorphisms of vector spaces or generally modules as well as ( Abelian ) groups are precisely the surjective homomorphisms.

Ring epimorphisms are generally not surjective, see below.

In the categories , the epimorphisms are precisely the extreme epimorphisms, namely the surjective morphisms. ${\displaystyle Set}$ ${\displaystyle Grp}$

In the category of topological spaces, the epimorphisms are the surjective continuous maps and the extreme epimorphisms are the quotient maps .

In the category of Hausdorff spaces the extremal epimorphisms are the same as in , but the epimorphisms are the continuous maps with dense image . This fact is often used in so-called “density conclusions”: To show that two continuous functions with a common domain dom (a Hausdorff space) are the same, it is sufficient to show that they agree on a dense subset of the domain. The inclusion mapping is an epimorphism, from which the equality follows over the entire domain of definition. ${\displaystyle Top_{2}}$ ${\displaystyle Top}$ ${\displaystyle D}$ ${\displaystyle D\to dom}$

In the category , the epimorphisms are the linear continuous mappings with a dense image (Banach spaces are Hausdorffsch) and the extreme epimorphisms are the surjective continuous linear mappings. ${\displaystyle BanSp_{1}}$

Epimorphism in Universal Algebra

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

Examples

If a homomorphism is, then it is surjective, i.e. an epimorphism. ${\displaystyle f\colon A\to B}$ ${\displaystyle f'\colon A\to \mathrm {im} \,f,a\mapsto f(a)}$

For every normal divisor of a group there is a canonical epimorphism that maps an element of to its remainder class . ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle p\colon G\to G/N}$ ${\displaystyle g}$ ${\displaystyle G}$ ${\displaystyle gN}$

The best-known examples of canonical epimorphisms are the mappings that assign the remainder of an integer when divided by a natural number , this remainder being understood as an element of the remainder class ring . ${\displaystyle m}$ ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$

The parallel projection is in linear algebra is a vector space -homomorphism that a vector space surjektiv a subspace maps.

Example of a non-surjective monoid epimorphism '

Consider the embedding morphism of the natural numbers including the zero in the whole numbers (both are monoids with addition as a link and as a neutral element): ${\displaystyle i}$ ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle +}$ ${\displaystyle 0}$

${\displaystyle i\colon \quad \mathbb {N} _{0}\to \mathbb {Z} ,\quad n\mapsto n}$.

It is not surjective and therefore not an epimorphism in the sense of universal algebra. However, it is an epimorphism in the monoid category.

Proof: Let it be a monoid with the operation and the neutral element . Let two otherwise arbitrary monoid homomorphisms with . It is to be shown that on whole ${\displaystyle M}$ ${\displaystyle +_{_{\!M}}}$ ${\displaystyle 0_{_{M}}}$ ${\displaystyle g,h\colon \,\mathbb {Z} \to M}$ ${\displaystyle g\circ i=h\circ i.}$ ${\displaystyle g=h}$ ${\displaystyle \mathbb {Z} .}$

Since, restricted to the non-negative integers, is reversible (and the identity ), there and agree. That they also agree on the negative numbers is shown by the following chain of equations, which applies to any negative number ( a notation for the additive inverse of so that is then positive): ${\displaystyle i}$ ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle z\in \mathbb {Z} }$ ${\displaystyle {\bar {z}}:=-z}$ ${\displaystyle z,}$ ${\displaystyle {\bar {z}}}$

${\displaystyle g(z)}$	${\displaystyle =}$	${\displaystyle g(z)+_{_{\!M}}0_{_{M}}}$	Definition of ${\displaystyle 0_{_{M}}\in M}$
	${\displaystyle =}$	${\displaystyle g(z)+_{_{\!M}}h(0)}$	${\displaystyle h}$ is monoid homomorphism
	${\displaystyle =}$	${\displaystyle g(z)+_{_{\!M}}h({\bar {z}}+z)}$	Property in ${\displaystyle \mathbb {Z} }$
	${\displaystyle =}$	${\displaystyle g(z)+_{_{\!M}}h({\bar {z}})+_{_{\!M}}h(z)}$	${\displaystyle h}$ is monoid homomorphism
	${\displaystyle =}$	${\displaystyle g(z)+_{_{\!M}}g({\bar {z}})+_{_{\!M}}h(z)}$	${\displaystyle g,h}$ agree on the positive numbers
	${\displaystyle =}$	${\displaystyle g(z+{\bar {z}})+_{_{\!M}}h(z)}$	${\displaystyle g}$ is monoid homomorphism
	${\displaystyle =}$	${\displaystyle g(0)+_{_{\!M}}h(z)}$	Property in ${\displaystyle \mathbb {Z} }$
	${\displaystyle =}$	${\displaystyle 0_{_{M}}+_{_{\!M}}h(z)}$	${\displaystyle g}$ is monoid homomorphism
	${\displaystyle =}$	${\displaystyle h(z)}$	Definition of ${\displaystyle 0_{_{M}}\in M}$

Thus, on the whole domain , ie an epimorphism.     ${\displaystyle g=h}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle i}$ ${\displaystyle \square }$

Incidentally, the much stronger statement applies:
If two monoid homomorphisms match on two consecutive numbers, then they match at all. ${\displaystyle g,h\colon \,\mathbb {Z} \to M}$

See also

• Monomorphism
• Isomorphism
• Homomorphism theorem

Individual evidence

1. ↑ Steve Awodey: Category theory . Clarendon Press, Oxford 2010, ISBN 0-19-923718-2 , pp. 25 .