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
The dual concept is that of coequalizer.
In type theory the equalizer
is given by the dependent sum over the dependent equality type
In $C =$ Set the equalizer of two functions of sets is the subset of elements of $c$ on which both functions coincide.
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 $f$.
For $S \stackrel{\overset{g}{\longrightarrow}}{\underset{f}{\longrightarrow}} T$ the given diagram, form the pullback along the diagonal morphism of $T$:
One checks that the horizontal morphism $eq(f,g) \to S$ equalizes $f$ and $g$ and that it does so universally.
If a category has equalizers and (finite) products, then it has (finite) limits.
For the finite case, we may say equivalently:
If a category has equalizers, binary products and a terminal object, then it has finite limits.
(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.
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.
Textbook account:
Last revised on September 4, 2021 at 12:01:07. See the history of this page for a list of all contributions to it.