Subsection 4.2.5 (00AS): Equalisers

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Style 4	`book`	Computer Modern	`amsthm`

The equaliser of $(f, g)$ is the pair consisting of:

The Limit. The pointed set $(Eq (f, g), x_{0})$ .
The Cone. The morphism of pointed sets

$eq (f, g) : (Eq (f, g), x_{0}) ↪ (X, x_{0})$

given by the canonical inclusion $eq (f, g) ↪ Eq (f, g) ↪ X$ .

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

f \circ eq (f, g) = g \circ eq (f, g),

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

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

ϕ : (E, *) \to (Eq (f, g), x_{0})

making the diagram

commute, being uniquely determined by the condition

eq (f, g) \circ ϕ = e

via

ϕ (x) = e (x)

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

f \circ e = g \circ e,

which gives

f (e (x)) = g (e (x))

for each $x \in E$ , so that $e (x) \in Eq (f, g)$ . Lastly, we note that $ϕ$ is indeed a morphism of pointed sets, as we have

\begin{aligned} ϕ (*) & = e (*) \\ = x_{0}, \end{aligned}

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

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

1. Associativity. We have isomorphisms of pointed sets

$\underset{= Eq (f \circ eq (g, h), h \circ eq (g, h))}{\underset{⏟}{Eq (f \circ eq (g, h), g \circ eq (g, h))}} ≅ Eq (f, g, h) ≅ \underset{= Eq (g \circ eq (f, g), h \circ eq (f, g))}{\underset{⏟}{Eq (f \circ eq (f, g), h \circ eq (f, g))}},$

where $Eq (f, g, h)$ is the limit of the diagram

in ${Sets}_{*}$ , being explicitly given by

$Eq (f, g, h) ≅ {a \in A | f (a) = g (a) = h (a)} .$
2. Unitality. We have an isomorphism of pointed sets
$Eq (f, f) ≅ X .$
3. Commutativity. We have an isomorphism of pointed sets
$Eq (f, g) ≅ Eq (g, f) .$

The Clowder Project

4.2.5 Equalisers