View source for Equalizer

{{TEX|done}}{{MSC|18A30}}

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 [[projective limit|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}}}