Quotient object (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 $u:Q\to Q^{\prime }$ satisfies $q^{\prime }=u\circ q$. 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.