= Quotient object
{disambiguate=category theory}
A quotient object of an <object of a category> $X$ is represented by an <epimorphism> $q:X\to Q$. Two such arrows represent the same quotient if an <isomorphism> $u:Q\to Q'$ satisfies $q'=u\circ q$. A quotient is therefore an arrow out of $X$, considered up to this equivalence, rather than an arbitrary smaller-looking object.
In the <Category of sets>, this recovers surjective maps and <quotient sets> by <equivalence relations>. In the <category of groups>, <quotient groups> $G/N$ arise from <normal subgroups>. Quotients can be ordered by factorization: $q_2=h\circ q_1$ means that $q_2$ identifies at least as much as $q_1$.
General <epimorphisms> need not be surjective maps of underlying sets. In the <category of commutative rings>, $\mathbb Z\hookrightarrow\mathbb Q$ is an <epimorphism>: two unital ring maps out of $\mathbb Q$ agreeing on $\mathbb Z$ agree on inverses and hence on every rational. Some uses therefore restrict quotient objects to <regular epimorphisms>, namely arrows realized as <coequalizers>; the intended convention should be stated.
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