Monomorphism

Monomorphism (from the Greek μόνος monos "one, alone" and μορφή morphé "shape, form") is a term from the mathematical sub-areas of algebra and category theory . In algebra, it denotes a homomorphism that is injective . In category theory, he generalizes the concept of injective mapping and allows objects to be viewed as sub-objects of others.

Note that universal algebra and category theory each explain a concept that is dual to monomorphism , namely epimorphism , but these two epimorphism concepts are not equivalent.

Monomorphisms of algebraic structures

A homomorphism of

• Vector spaces or more general modules
• or ( Abelian ) groups
• or wrestling or bodies
• or general algebraic structures ,

which is injective is called monomorphism .

Examples

• The mapping with is a vector space monomorphism. ${\displaystyle \operatorname {h} \colon \mathbb {R} ^{2}\to \mathbb {R} ^{3}}$ ${\displaystyle \operatorname {h} \left(x,y\right)=\left(x,y,x+y\right)}$
• The mapping with is a group homomorphism , but not injective. ${\displaystyle \operatorname {g} \colon \left(\mathbb {C} ,+\right)\to \left(\mathbb {R} ,+\right)}$ ${\displaystyle \operatorname {g} \left(z\right)=\operatorname {Re} \left(z\right)}$
• A homomorphism of groups, rings or modules (especially vector spaces) is injective if and only if its kernel is trivial. For any homomorphism of groups, rings or modules (or vector spaces) there is a monomorphism if the canonical mapping is on the remainder class structure. Because it applies and therefore is trivial. ${\displaystyle \operatorname {f} \colon A\to B}$
       ${\displaystyle {\tilde {\operatorname {f} }}\colon A/{\ker {\operatorname {f} }}\to B}$
  ${\displaystyle \operatorname {\tilde {f}} \colon [a]\mapsto \operatorname {f} (a)}$ ${\displaystyle \ker \operatorname {\tilde {f}} =\lbrace \ker \operatorname {f} \rbrace }$ ${\displaystyle \ker \operatorname {\tilde {f}} }$
• Homomorphisms of bodies are always injective, i.e. always monomorphisms.

Monomorphisms of relational structures

For more general structures (in the sense of model theory), especially for relational structures, a monomorphism is defined as an injective strong homomorphism . Equivalent to this: The mapping is an isomorphism on its image. The above definition is obtained for the special case of algebraic structures, since every homomorphism between algebraic structures is strong.

Monomorphisms in any categories

definition

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

If any morphisms are with , then it follows (One also says: can be shortened on the left ). ${\displaystyle g,h\colon T\to X}$ ${\displaystyle f\circ g=f\circ h}$ ${\displaystyle g=h}$ ${\displaystyle f}$

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

In categories of algebraic structures as well as in the categories of sets or topological spaces, the monomorphisms are precisely the injective morphisms. But there are also concrete categories with non-injective monomorphisms.

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

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

or using a hook arrow with two terms as

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

written down.

Example of a non-injective monomorphism

We consider the category Div of the divisible groups : The objects are the Abelian groups for which the following applies: ${\displaystyle G}$

For all and all , a exist with ; the element can therefore be “divided by ”. ${\displaystyle a\in G}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle n>0}$ ${\displaystyle b\in G}$ ${\displaystyle a=nb}$ ${\displaystyle a}$ ${\displaystyle n}$

The morphisms are the group homomorphisms between these groups.

The Abelian groups and are in this category. The canonical projection is surjective but not injective. We show that it is a monomorphism in Div . ${\displaystyle (\mathbb {Q} ,+)}$ ${\displaystyle (\mathbb {Q} /\mathbb {Z} ,+)}$ ${\displaystyle \pi \colon \mathbb {Q} \to \mathbb {Q} /\mathbb {Z} }$

If namely is any divisible group and there are two morphisms with the property , then applies . Well then there would be a with . When you swap the roles of and , so that you definitely get. Since it is divisible, there would then be a with . But then it would be ${\displaystyle X}$ ${\displaystyle a,b\colon X\to \mathbb {Q} }$ ${\displaystyle \pi \circ a=\pi \circ b}$ ${\displaystyle \forall x\in X:a(x)-b(x)\in \operatorname {ker} \pi =\mathbb {Z} }$ ${\displaystyle a\neq b}$ ${\displaystyle x\in X}$ ${\displaystyle t:=a(x)-b(x)\neq 0}$ ${\displaystyle t<0}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle t\in \mathbb {N} }$ ${\displaystyle X}$ ${\displaystyle y\in X}$ ${\displaystyle x=2t\;y}$

${\displaystyle t=a(x)-b(x)=a(2t\;y)-b(2t\;y)=2t(a(y)-b(y))}$,

so what would contradict. ${\displaystyle a(y)-b(y)=1/2}$ ${\displaystyle \forall x:a(x)-b(x)\in \mathbb {Z} }$

Extremal monomorphisms

→ Main article : Extreme monomorphism and epimorphism

A monomorphism is called extremal if it also fulfills the following extremal property: ${\displaystyle f}$

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

Because there is automatically a monomorphism, in categories in which all bimorphisms (that is, monomorphisms that are epimorphisms) are already isomorphisms, all monomorphisms are extremal. One has this, for example, in the category of sets and the category of groups. ${\displaystyle m}$

In the category of topological spaces, the extremal monomorphisms are the embeddings . In the category of Hausdorff spaces , the extremal monomorphisms are the closed embeddings.

In the category of Banach spaces , the extremal monomorphisms are precisely those linear continuous injective mappings for which there is a positive one , so that for all of the domain of definition: ${\displaystyle f}$ ${\displaystyle m}$ ${\displaystyle x}$

${\displaystyle m\|x\|\leq \|f(x)\|}$

Sub-objects

For a given object of a category , one can consider the sub-category of the disk category , the objects of which are all monomorphisms in . Parallel arrows are always identical here; so it is a quasi-order . The partial order of the sub-objects of is now the one that arises from the transition to isomorphism classes. ${\displaystyle A}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {QSub} (A)}$ ${\displaystyle {\mathcal {C}}/A}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathrm {Sub} (A)}$ ${\displaystyle A}$ ${\displaystyle \mathrm {QSub} (A)}$

See also

• Isomorphism

Individual evidence

1. ↑ Philipp Rothmaler: Introduction to the model theory . Spektrum Akademischer Verlag, 1995, ISBN 978-3-86025-461-5 , pp. 21 .
2. ↑ Steve Awodey: Category theory . Clarendon Press, Oxford 2010, ISBN 0-19-923718-2 , pp. 25 .