Monomorphism

Monomorphism (from the Greek monos μόνος one, alone and Greek μορφή morphé shape, form ) is a term used in the mathematical sciences of algebra and category theory. In algebra he called a homomorphism, which is injective. In category theory, he generalizes the notion of injective map and allows objects as child objects of other limiting.

Note that the universal algebra and category theory in each case a dual to monomorphism concept, that epimorphism, explain these two epimorphism terms but are not equivalent.

• 3.1 Definition
• 3.2 Example of a non- injective monomorphism
• 3.3 Extremal monomorphisms

Monomorphisms of algebraic structures

A homomorphism of

• Vector spaces, or more generally modules
• Or ( abelian ) groups
• Or rings or bodies
• Or general algebraic structures,

Is injective, ie monomorphism.

Examples

• The figure with is a vector space monomorphism.
• The figure with is indeed a group homomorphism, but not injective.
• A homomorphism of groups, rings and modules ( especially vector spaces ) is injective if its kernel is trivial. For an arbitrary homomorphism of groups, rings and modules ( or vector spaces ) is a monomorphism if the canonical map on the residue class structure. For it is true and so is trivial.
• Homomorphisms of bodies are always injective, so 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 homomorphism strong. Equivalently: The illustration is an isomorphism onto its image. For the special case of algebraic structures is obtained, the above definition, since every homomorphism between algebraic structures is strong.

Monomorphisms in any of the categories

Definition

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

(together with ) then that means a sub-object of.

Categories of algebraic structures as well as in the categories of quantity, or topological spaces, the monomorphisms are exactly the injective morphisms. But there are also specific categories with non- injective monomorphisms.

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

Or by using a hook arrow 2 terms as

Noted.

Example of a non- injective monomorphism

We consider the category Div of divisible groups: The objects are the abelian groups G, applies to the following:

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. Indeed, if X is any divisible group and two morphisms with the property, then apply. Now if, then there would be a with. Swap the rollers in one of a and b, so as to obtain in any case. Since X is separable, then there would be with a. But then would

So, which would contradict.

Extremal monomorphisms

A monomorphism is called extremal if it satisfies the following additional extremal:

In the categories of quantities or groups of extremal monomorphisms are precisely the monomorphisms.

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

In the category of Banach spaces the extremal monomorphisms are precisely those linear continuous injective images, for which there is a positive such that for all x in the domain of definition: