where G is some divisible group, and 1=''q'' ? ''h'' = 0, then ''h''(''x'') ? '''Z''', ? ''x'' ? ''G''. Now fix some ''x'' ? ''G''. Without loss of generality, we may assume that ''h''(''x'') >= 0 (otherwise, choose -x instead). Then, letting 1=''n'' = ''h''(''x'') + 1, since G is a divisible group, there exists some ''y'' ? ''G'' such that 1=''x'' = ''ny'', so 1=''h''(''x'') = ''n'' ''h''(''y''). From this, and 1=0 <= ''h''(''x'') < ''h''(''x'') + 1 = ''n'', it follows that

Since ''h''(''y'') ? '''Z''', it follows that 1=''h''(''y'') = 0, and thus 1=''h''(''x'') = 0 = ''h''(-''x''), ? ''x'' ? ''G''. This says that 1=''h'' = 0, as desired.

To go from that implication to the fact that q is a monomorphism, assume that 1=''q'' ? ''f'' = ''q'' ? ''g'' for some morphisms ''f'', ''g'' : ''G'' -> '''Q''', where G is some divisible group. Then 1=''q'' ? (''f'' - ''g'') = 0, where (''f'' - ''g'') : ''x'' ? ''f''(''x'') - ''g''(''x''). (Since 1=(''f'' - ''g'')(0) = 0, and 1=(''f'' - ''g'')(''x'' + ''y'') = (''f'' - ''g'')(''x'') + (''f'' - ''g'')(''y''), it follows that (''f'' - ''g'') ? Hom(''G'', '''Q''')). From the implication just proved, 1=''q'' ? (''f'' - ''g'') = 0 => ''f'' - ''g'' = 0 <=> ? ''x'' ? ''G'', ''f''(''x'') = ''g''(''x'') <=> ''f'' = ''g''. Hence q is a monomorphism, as claimed.

Properties

*In a topos, every mono is an equalizer, and any map that is both monic and epic is an isomorphism.
*Every isomorphism is monic.

Related concepts

There are also useful concepts of regular monomorphism, extremal monomorphism, immediate monomorphism, strong monomorphism, and split monomorphism.

* A monomorphism is said to be regular if it is an equalizer of some pair of parallel morphisms.
* A monomorphism is said to be extremal if in each representation , where is an epimorphism, the morphism is automatically an isomorphism.
* A monomorphism is said to be immediate if in each representation , where is a monomorphism and is an epimorphism, the morphism is automatically an isomorphism.
* A monomorphism is said to be strong if for any epimorphism and any morphisms and such that , there exists a morphism such that and .
* A monomorphism is said to be split if there exists a morphism such that (in this case is called a left-sided inverse for ).

Terminology

The companion terms monomorphism and epimorphism were originally introduced by Nicolas 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 cancellation 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.

In adjectival form, a monomorphism is said to be monic; in common shorthand, it is also called a mono.