Equalizer

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

Definition

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

1. (Equalizing condition) $f\circ e = g\circ e$,
2. (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 $e\circ u = h.$

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

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

Basic 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

1. $\mathbf{Set}$.
   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.

2. $\mathbf{Grp}$.
   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 $G$.

3. $\mathbf{Ab}$ (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 $E\hookrightarrow M$.