nLab equalizer

Definition

In category theory

An equalizer is a limit

$$\operatorname{eq}\underset{\quad e \quad}{\to}x\underoverset{\quad g \quad}{f}{\rightrightarrows}y$$

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

$$\left\lbrace x \underoverset{\quad g \quad}{f}{\rightrightarrows} y \right\rbrace.$$

(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

$$P \to A \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} B$$

is given by the dependent sum over the dependent equality type

$$P \simeq \sum_{a : A} (f(a) = g(a)).$$

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

Proposition

A category has equalizers if it has binary products and pullbacks.

Proof

For $S \stackrel{\overset{g}{\longrightarrow}}{\underset{f}{\longrightarrow}} T$ the given diagram, form the pullback along the diagonal morphism of $T$:

$$\array{ eq(f,g) &\longrightarrow& S \\ \big\downarrow && \big\downarrow {}^{\mathrlap{(f, g)}} \\ T &\underset{(id, id)}{\longrightarrow}& T \times T }.$$

One checks that the horizontal morphism $eq(f,g) \to S$ equalizes $f$ and $g$ and that it does so universally.

Proposition

If a category has equalizers and (finite) products, then it has (finite) limits.

For the finite case, we may say equivalently:

Proposition

If a category has equalizers, binary products and a terminal object, then it has finite limits.

Proposition

(Eckmann and Hilton EH, Proposition 1.3.) Let $e: E \rightarrow X$ be an arrow in a category $\mathcal{C}$ which is an equaliser of a pair of arrows of $\mathcal{C}$. Then $e$ is a monomorphism.

Proof

If $g,h : A \rightarrow E$ are arrows of $\mathcal{C}$ such that $e \circ g = e \circ h$, then it follows immediately from the uniqueness part of the universal property of an equaliser that $g = h$.

Proposition

$f = g$ if and only if $id : X \to X$ is an equalizer of $f$ and $g$. Similarly, given any equalizer $e : eq \to X$ of $f$ and $g$, $f = g$ if and only if $e$ is an isomorphism, which because $e$ is monic, one only needs to verify that it has a section.

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)