# nLab equalizer

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## 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

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

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 $g = h$.

## References

Equalizers were defined in the paper

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

Last revised on March 27, 2021 at 11:57:04. See the history of this page for a list of all contributions to it.