Contents

Definition

In category theory

An equalizer is a limit

over a parallel pair i.e. of the diagram of the shape

(See also fork diagram).

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

• an object $eq(f,g) \in C$;

• a morphism $eq(f,g) \to x$

• such that

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

The dual concept is that of coequalizer.

In type theory

In type theory the equalizer

is given by the dependent sum over the dependent equality type

Examples

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

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

• 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 $f$.

Properties

For the finite case, we may say equivalently:

Related concepts

• homotopy equalizer

References

Equalizers were defined in the paper

• Beno Eckmann, Peter J. Hilton, Group-like structures in general categories II. Equalizers, limits, lengths. Mathematische Annalen 151:2 (1963), 150–186. doi:10.1007/bf01344176.

for any finite collection of parallel morphisms. The paper refers to them as left equalizers, whereas coequalizers are referred to as right equalizers.

Textbook account:

• Francis Borceux, Section 2.4 in Vol. 1: Basic Category Theory of: Handbook of Categorical Algebra, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)