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In the context of
abstract algebra or
universal algebra, a '''monomorphism''' is simply an
injective homomorphism.
In the more general (and abstract) setting of
category theory, a '''monomorphism''' (also called a '''monic morphism''') 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''.
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).
In the the
category of sets the monomorphisms are exactly the
injective morphisms. Thus the algebraic and categorical notions are the same. The same is true in many other
concrete categories such as those of
groups,
rings, and
vector spaces. (''Are there any counterexamples?'')
''See also'':
*
epimorphism
*
isomorphism
*
subobject
