Equalizer - Knowlpedia

Let $\mathcal C$ be a category and let $f,g:A\to B$ be parallel morphisms in $\mathcal C$ .

An equalizer of $f$ and $g$ is a morphism $e:E\to A$ such that:

(Equalizing condition) $f\circ e = g\circ e$ ,
(Universal property) for any morphism $h:X\to A$ with $f\circ h=g\circ h$ , there exists a unique morphism $u:X\to E$ such that $e\circ u = h.$

Equivalent characterizations

Equivalently, $e:E\to A$ is universal among arrows into $A$ on which $f$ and $g$ agree:

with $f\circ e=g\circ e$ and $f\circ h=g\circ h$ .

Remarks

An equalizer (when it exists) is a special case of a limit.

Properties

The equalizer morphism $e:E\to A$ is always a monomorphism: it is “injective” in the categorical sense.
In an abelian category, the equalizer of $f,g$ can be identified with a kernel: $\mathrm{Eq}(f,g)\;\cong\;\ker(f-g).$

Examples

$\mathbf{Set}$ . If $f,g:A\to B$ are functions, the equalizer is the subset $E=\{a\in A \mid f(a)=g(a)\}\subseteq A,$ with $e:E\hookrightarrow A$ the inclusion.

$\mathbf{Grp}$ . For homomorphisms $f,g:G\to H$ , the equalizer is the subgroup $E=\{x\in G \mid f(x)=g(x)\}\le G,$ included into $G$ .

$\mathbf{Ab}$ (or $R$ -Mod). For module homomorphisms $f,g:M\to N$ , the equalizer is the submodule $E=\ker(f-g)\subseteq M,$ with the inclusion $E\hookrightarrow M$ .