nLab
equalizer

Definition

An equalizer is a limit over a diagram of the shape

$$\left\{a\overrightarrow{\to }b\right\}.$$\left\lbrace a \stackrel{\to}{\to} b \right\rbrace \,.

This means that for $f:c\to d$ and $g:c\to d$ two parallel morphisms in a category $C$, their equalizer is, if it exists

• an object $\mathrm{Eq}(f,g)\in C$;

• a morphism $\mathrm{Eq}(f,g)\to c$

• such that

  • pulled back to $\mathrm{Eq}(f,g)$ both morphisms become equal: $(\mathrm{Eq}(f,g)\to c\stackrel{f}{\to }d)=(\mathrm{Eq}(f,g)\to c\stackrel{g}{\to }d)$
  • and $\mathrm{Eq}(f,g)$ is the universal object with this property.

Examples

• In $C=$ Set the equalizer of two functions of sets is the subset of elements of $c$ on which both functions coincide.

  $$\mathrm{Eq}(f,g)=\{s\in c\mid f(s)=g(s)\}.$$Eq(f,g) = \left\{ s \in c | f(s) = g(s) \right\} \,.

• For $C$ a category with zero object the equalizer of a morphism $f:c\to d$ with the corresponding zero morphism is the kernel of $f$.