Context
Limits and colimits
limits and colimits
1Categorical

limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit, wide pullback

preserved limit, reflected limit, created limit

product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum

finite limit

Kan extension

weighted limit

end and coend
2Categorical
(∞,1)Categorical
Modelcategorical
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
$\left\lbrace
x \underoverset{\quad g \quad}{f}{\rightrightarrows} y
\right\rbrace
\,.$
(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
Proof
For $S \stackrel{\overset{g}{\longrightarrow}}{\underset{f}{\longrightarrow}} T$ the given diagram, first form the pullback
$\array{
S \times_{f,g} S &\to& S
\\
\downarrow && \downarrow^{\mathrlap{g}}
\\
S &\stackrel{f}{\to}& T
}
\,.$
This gives a morphism $S \times_{f,g} S \to S \times S$ into the product.
Define $eq(f,g)$ to be the further pullback
$\array{
eq(f,g) &\to& S \times_{f,g} S
\\
\downarrow && \downarrow
\\
S &\stackrel{(id, id)}{\to}& S \times S
}
\,.$
One checks that the vertical morphism $eq(f,g) \to S$ equalizes $f$ and $g$ and that it does so universally.
Proposition
If a category has products and equalizers, then it has limits.