Equalizer

2020 Mathematics Subject Classification: Primary: 18A30 [MSN][ZBL]

An equaliser of two morphisms $f,g$ between the objects $X, Y$ of a category $\mathfrak{K}$ is a morphism $e : W \rightarrow X$ such that $ef = eh$ and any morphism $d : A \rightarrow X$ such that $df = dg$ factors through $e$, that is, there exists $c : A \rightarrow W$ such that $cd = e$. A coequaliser is the dual notion.

An equaliser in the category of sets exists: it is the inclusion map on $\{ x \in X : f(x) = g(x) \}$. Similarly, a co-equaliser exists: it is the quotient map on $X$ determined by the equivalence relation $\sim$ generated by $f(x) \sim g(x),\ x \in X$.

If $\mathfrak{J}$ is the category ${\downarrow}{\downarrow}$, and $F$ is a functor from $\mathfrak{J}$ to $\mathfrak{K}$, then a limit of $F$ is an equaliser and a colimit of $F$ is a coequaliser.

Every equaliser in a category $\mathfrak{K}$ is a monomorphism and every coequaliser is an epimorphism. A monomorphism (resp. epimorphism) which is an equaliser (resp. coequaliser) is termed regular.

References

• Saunders MacLane, "Categories for the working mathematician" Graduate Texts in Mathematics 5 Springer (1988) ISBN 0-387-98403-8 Zbl 0705.18001