## 4.6 Fibre products

Definition 4.6.1. Let $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(z, y)$. A *fibre product* of $f$ and $g$ is an object $x \times _ y z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ y z, x)$ and $q \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ y z, z)$ making the diagram

\[ \xymatrix{ x \times _ y z \ar[r]_ q \ar[d]_ p & z \ar[d]^ g \\ x \ar[r]^ f & y } \]

commute, and such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(w, z)$ with $f \circ \alpha = g \circ \beta $ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x \times _ y z)$ making the diagram

\[ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times _ y z \ar[d]^ p \ar[r]_ q & z \ar[d]^ g \\ & & x \ar[r]^ f & y } \]

commute.

If a fibre product exists it is unique up to unique isomorphism. This follows from the Yoneda lemma as the definition requires $x \times _ y z$ to be an object of $\mathcal{C}$ such that

\[ h_{x \times _ y z}(w) = h_ x(w) \times _{h_ y(w)} h_ z(w) \]

functorially in $w$. In other words the fibre product $x \times _ y z$ is an object representing the functor $w \mapsto h_ x(w) \times _{h_ y(w)} h_ z(w)$.

Definition 4.6.2. We say a commutative diagram

\[ \xymatrix{ w \ar[r] \ar[d] & z \ar[d] \\ x \ar[r] & y } \]

in a category is *cartesian* if $w$ and the morphisms $w \to x$ and $w \to z$ form a fibre product of the morphisms $x \to y$ and $z \to y$.

Definition 4.6.3. We say the category $\mathcal{C}$ *has fibre products* if the fibre product exists for any $f\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x, y)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(z, y)$.

Definition 4.6.4. A morphism $f : x \to y$ of a category $\mathcal{C}$ is said to be *representable* if for every morphism $z \to y$ in $\mathcal{C}$ the fibre product $x \times _ y z$ exists.

Lemma 4.6.5. Let $\mathcal{C}$ be a category. Let $f : x \to y$, and $g : y \to z$ be representable. Then $g \circ f : x \to z$ is representable.

**Proof.**
Omitted.
$\square$

Lemma 4.6.6. Let $\mathcal{C}$ be a category. Let $f : x \to y$ be representable. Let $y' \to y$ be a morphism of $\mathcal{C}$. Then the morphism $x' := x \times _ y y' \to y'$ is representable also.

**Proof.**
Let $z \to y'$ be a morphism. The fibre product $x' \times _{y'} z$ is supposed to represent the functor

\begin{eqnarray*} w & \mapsto & h_{x'}(w)\times _{h_{y'}(w)} h_ z(w) \\ & = & (h_ x(w) \times _{h_ y(w)} h_{y'}(w)) \times _{h_{y'}(w)} h_ z(w) \\ & = & h_ x(w) \times _{h_ y(w)} h_ z(w) \end{eqnarray*}

which is representable by assumption.
$\square$

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