Editing Monomorphism (section)

==Relation to invertibility==
Left-invertible morphisms are necessarily monic: if ''l'' is a left inverse for ''f'' (meaning ''l'' is a morphism and <math>l \circ f = \operatorname{id}_{X}</math>), then ''f'' is monic, as
: <math>f \circ g_1 = f \circ g_2 \Rightarrow l\circ f\circ g_1 = l\circ f\circ g_2 \Rightarrow g_1 = g_2.</math>

A left-invertible morphism is called a '''[[Section (category theory)|split mono]]''' or a '''section'''. 

However, a monomorphism need not be left-invertible. For example, in the category '''Group''' of all [[Group (mathematics)|groups]] and [[group homomorphism]]s among them, if ''H'' is a subgroup of ''G'' then the inclusion {{nowrap|''f'' : ''H'' → ''G''}} is always a monomorphism; but ''f'' has a left inverse in the category if and only if ''H'' has a [[Complement (group theory)|normal complement]] in ''G''. 

A morphism {{nowrap|''f'' : ''X'' → ''Y''}} is monic if and only if the induced map {{nowrap|''f''<sub>∗</sub> : Hom(''Z'', ''X'') → Hom(''Z'', ''Y'')}}, defined by {{nowrap|1=''f''<sub>∗</sub>(''h'') = ''f'' ∘ ''h''}} for all morphisms {{nowrap|''h'' : ''Z'' → ''X''}}, is [[injective]] for all objects ''Z''.