Definition 6.2.5.1.1 (00AT): Equalisers of Pointed Sets

Concretely, the equaliser of $\webleft (f,g\webright )$ is the pair consisting of:

•
The Limit. The pointed set $\webleft (\operatorname {\mathrm{Eq}}\webleft (f,g\webright ),x_{0}\webright )$.
•
The Cone. The morphism of pointed sets

\[ \operatorname {\mathrm{eq}}\webleft (f,g\webright )\colon \webleft (\operatorname {\mathrm{Eq}}\webleft (f,g\webright ),x_{0}\webright )\hookrightarrow \webleft (X,x_{0}\webright ) \]

given by the canonical inclusion $\operatorname {\mathrm{eq}}\webleft (f,g\webright )\hookrightarrow \operatorname {\mathrm{Eq}}\webleft (f,g\webright )\hookrightarrow X$.

We claim that $\webleft (\operatorname {\mathrm{Eq}}\webleft (f,g\webright ),x_{0}\webright )$ is the categorical equaliser of $f$ and $g$ in $\mathsf{Sets}_{*}$. First we need to check that the relevant equaliser diagram commutes, i.e. that we have

\[ f\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright )=g\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright ), \]

which indeed holds by the definition of the set $\operatorname {\mathrm{Eq}}\webleft (f,g\webright )$. Next, we prove that $\operatorname {\mathrm{Eq}}\webleft (f,g\webright )$ satisfies the universal property of the equaliser. Suppose we have a diagram of the form

in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets

\[ \phi \colon \webleft (E,*\webright )\to \webleft (\operatorname {\mathrm{Eq}}\webleft (f,g\webright ),x_{0}\webright ) \]

making the diagram

commute, being uniquely determined by the condition

\[ \operatorname {\mathrm{eq}}\webleft (f,g\webright )\circ \phi =e \]

via

\[ \phi \webleft (x\webright )=e\webleft (x\webright ) \]

for each $x\in E$, where we note that $e\webleft (x\webright )\in A$ indeed lies in $\operatorname {\mathrm{Eq}}\webleft (f,g\webright )$ by the condition

\[ f\circ e=g\circ e, \]

which gives

\[ f\webleft (e\webleft (x\webright )\webright )=g\webleft (e\webleft (x\webright )\webright ) \]

for each $x\in E$, so that $e\webleft (x\webright )\in \operatorname {\mathrm{Eq}}\webleft (f,g\webright )$. Lastly, we note that $\phi $ is indeed a morphism of pointed sets, as we have

\begin{align*} \phi \webleft (*\webright ) & = e\webleft (*\webright )\\ & = x_{0},\end{align*}

where we have used that $e$ is a morphism of pointed sets.

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets and let $f,g,h\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$ be morphisms of pointed sets.

1.
Associativity. We have isomorphisms of pointed sets

\[ \underbrace{\operatorname {\mathrm{Eq}}\webleft (f\circ \operatorname {\mathrm{eq}}\webleft (g,h\webright ),g\circ \operatorname {\mathrm{eq}}\webleft (g,h\webright )\webright )}_{{}=\operatorname {\mathrm{Eq}}\webleft (f\circ \operatorname {\mathrm{eq}}\webleft (g,h\webright ),h\circ \operatorname {\mathrm{eq}}\webleft (g,h\webright )\webright )}\cong \operatorname {\mathrm{Eq}}\webleft (f,g,h\webright ) \cong \underbrace{\operatorname {\mathrm{Eq}}\webleft (f\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright ),h\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright )\webright )}_{{}=\operatorname {\mathrm{Eq}}\webleft (g\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright ),h\circ \operatorname {\mathrm{eq}}\webleft (f,g\webright )\webright )}, \]
where $\operatorname {\mathrm{Eq}}\webleft (f,g,h\webright )$ is the limit of the diagram
in $\mathsf{Sets}_{*}$, being explicitly given by

\[ \operatorname {\mathrm{Eq}}\webleft (f,g,h\webright )\cong \left\{ a\in A\ \middle |\ f\webleft (a\webright )=g\webleft (a\webright )=h\webleft (a\webright )\right\} . \]
2.
Unitality. We have an isomorphism of pointed sets

\[ \operatorname {\mathrm{Eq}}\webleft (f,f\webright )\cong X. \]
3.
Commutativity. We have an isomorphism of pointed sets

\[ \operatorname {\mathrm{Eq}}\webleft (f,g\webright ) \cong \operatorname {\mathrm{Eq}}\webleft (g,f\webright ). \]

The Clowder Project

6.2.5 Equalisers