= Subobject
{wiki}
A <subobject> of an object $A$ in a <category> is an equivalence class of <monomorphisms> $m:B\to A$. Two such maps are equivalent if an <isomorphism> between their domains makes the maps into $A$ agree. This removes dependence on the name of the chosen domain.
In the <Category of sets>, subobjects correspond to subsets; in the category of <groups>, they correspond to <subgroups>. The map into $A$ remains essential: two abstractly isomorphic subgroups can define different subobjects if they occupy different positions in $A$. Dually, <Quotient object (category theory)>[quotient objects] are equivalence classes of <epimorphisms> out of an object.
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