# 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 $g = h$.

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.

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