Coequalizer

or quotient object

Definition

• Given a morphism $f:A\to B$, the kernel pair $A{\times }_{B}A\rightrightarrows A$ defines an equivalence relation (a congruence in algebraic categories).
• The quotient $A/\sim$ is the coequalizer of those two morphisms.

$$A{\times }_{B}A\rightrightarrows A\to A/\sim$$

Example

• $G\to G/N$ in Grp (see Section 14 Factor Groups)
• $R\to R/I$ in Ring (see Section 26 Homomorphisms and Factor Rings)
• $S\to S/\sim$ in Set (see Equivalence Relations)