Information AboutMonomorphism |
| CATEGORIES ABOUT MONOMORPHISM | |
| abstract algebra | |
| category theory | |
| genetics | |
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In the context of Abstract Algebra or Universal Algebra , a monomorphism is simply an Injective Homomorphism . In the more general setting of Category Theory , a monomorphism (also called a '''monic morphism''' or a '''mono''') is a Morphism ''f'' : ''X'' → ''Y'' such that : ''f'' o ''g''1 = ''f'' o ''g''2 implies ''g''1 = ''g''2 for all morphisms ''g''1, ''g''2 : ''Z'' → ''X''. Monomorphisms are analogues of injective functions, but they are not exactly the same. The Dual of a monomorphism is an Epimorphism (i.e. a monomorphism in a category ''C'' is an epimorphism in the Dual Category ''C''op). TERMINOLOGY The companion terms ''monomorphism'' and '' Epimorphism '' were originally introduced by Bourbaki ; Bourbaki uses ''monomorphism'' as shorthand for an Injective Function . Early category theorists believed that the correct generalization of injectivity to the context of categories was the property given above. While this is not exactly true for monic maps, it is very close, so this has caused little trouble, unlike the case of epimorphisms. Saunders Mac Lane attempted to make a distinction between what he called ''monomorphisms'', which were maps in a concrete category whose underlying maps of sets were injective, and ''monic maps'', which are monomorphisms in the categorical sense of the word. This distinction never came into general use. EXAMPLES Every morphism in a Concrete Category whose underlying Function is Injective is a monomorphism. In the Category Of Sets , the converse also holds so the monomorphisms are exactly the Injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a Free Object on one generator. In particular, it is true in the categories of groups and rings, and in any Abelian Category . It is not true in general, however, that all monomorphisms must be injective in other categories. For example, in the category Div of ). This implies that ''h''(''x'') is an integer if ''x'' ∈ ''G''. If ''h''(''x'') is not 0 then, for instance, : so that :, contradicting ''q'' o ''h'' = 0, so ''h''(''x'') = 0 and ''q'' is therefore a monomorphism. RELATED CONCEPTS There are also useful concepts of regular monomorphism, '''strong monomorphism''', and '''extremal monomorphism'''. A regular monomorphism Equalizes some parallel pair of morphisms. An extremal monomorphism is a monomorphism that cannot be nontrivially factored through an epimorphism: Precisely, if ''m''=''g'' o ''e'' with ''e'' an epimorphism, then ''e'' is an isomorphism. A strong monomorphism satisfies a certain lifting property with respect to commutative squares involving an epimorphism. SEE ALSO REFERENCES
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