Monomorphism

Within the framework of the general Algebra or universal Algebra, a monomorphism is simply a injective Homomorphisme .

Within the more general framework of the Theory of the categories, a monomorphism (also called mono ) is a Morphisme that can be simplified on the left, i.e. an application $f\; \backslash \; colonist\; X\; \backslash \; to\; Y$ such as

$f\; \backslash \; circ\; g\_1\; =\; F\; \backslash \; circ\; g\_2\; \backslash \; implies\; g\_1\; =\; g\_2$ for any morphism $g\_1,\; g\_2\; \backslash \; colonist\; Z\; \backslash \; to\; X$.

Monomorphisms are generalization with the categories of the injective functions; in certain categories, the two concepts coincide besides. But monomorphisms remain more general objects (see the example below).

The dual of a monomorphism is a epimorphism (i.e. a monomorphism in the category C is a epimorphism in the dual category C ^op).

Terminology

The devoted terms monomorphism and epimorphism were originally introduced by Bourbaki; Bourbaki used monomorphism as short cut to indicate the injective functions. Later, the teoricians of the categories thought that the correct generalization of injectivity to the cases of the categories was that given by Bourbaki. Even if it is not exactly true for the case of the applications of the monomorphism type, which because some misunderstandings, contrary to the cases of the epimorphisms. Saunders Mac Lane tried to make the distinction between what it calls monomorphisms , which are concrete applications and whose subjacent applications on units are injective, and the applications monics , which are monomorphisms with the clean direction of the categories. However, this distinction never spread.

Inversibility

The invertible applications on the left are necessarily mono: if L is on the left the reverse of F (i.e. $lf=id\_X$), then F is mono, because

$f\; \backslash \; circ\; g\_1\; =\; F\; \backslash \; circ\; g\_2\; \backslash \; implies\; lfg\_1\; =\; lfg\_2\; \backslash \; implies\; g\_1\; =\; g\_2$

A iversible application on the left is called a ''' Split mono '''.

An application $f\; \backslash \; colonist\; X\; \backslash \; to\; Y$ is mono if and only if the application $f\_*\; \backslash \; colonist\; \backslash \; operatorname\; \{Hom\}\; (Z,\; X)\; \backslash \; to\; \backslash \; operatorname\; \{Hom\}\; (Z,\; Y)$ injective for all is Z .

Examples

Any morphism of a Catégorie concretes whose subjacent functions are Injective S is a monomorphism. In a Category of units, the reciprocal one is true, and thus monomorphisms are exactly the injective morphisms. The reciprocal one is also true in the majority of the usual categories from the existence of free objects on a generator. In particular, it is true for the categories of groups and rings, and for all abelian Catégorie.

On the other hand, all monomorphisms are necessarily not injective on other categories. For example, in the category Div of the abelian groups divisible and of the Homomorphism of groups between them, there are monomorphisms which are not injective: to consider the application quotient Q  :   Q   →   Q / Z . It is not injective; however, it is a monomorphism of this category. To see it, note that if Q   o  F = Q   o  G for a morphism F , G  :   G → Q where G is a divisible abelian group then Q   o  H = 0 where H = F - G (what has a direction in a additive Catégorie). That implies that H ( X ) is whole if X ∈ G . If H ( X ) is not null then,

$h\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{4h\; (X)\}\backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{4\}$

thus

$(Q\; \backslash \; circ\; H)\; \backslash \; left\; (\backslash \; frac\; \{X\}\; \{4h\; (X)\}\backslash \; right)\; \backslash \; neq\; 0$,

contradiction with Q   o  H = 0, therefore H ( X ) = 0 and Q is thus a monomorphism.

Dependant concepts

There exist also the concepts of regular monomorphism , strong monomorphism , and monomorphism extreme . A regular monomorphism equalizes a couple of morphisms. An extreme monomorphism is a monomorphism which cannot be triviallement factorized using a epimorphism: more precisely, if m = G   o  E with E a epimorphism, then E is an isomorphism. A monomorphism checks certain properties compared to commutative diagram implying epimorphisms.

See too

• Injection (mathematics)
• epimorphism
• Isomorphism

Refer

• Francis Borceaux (1994), Handbook off Categorical Algebra 1 , Cambridge University Near. ISBN 0-521-44178-1.
• George Bergman (1998), Year Invitation to General Algebra and Universal Constructions , Henry Helson Publisher, Berkeley. ISBN 0-9655211-4-1.
• Jaap van Oosten, BASIC Category Theory

Sources

Category: theory of the categories

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