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14.7 Fibre products of simplicial objects

Of course we should define the fibre product of simplicial objects as the fibre product in the category of simplicial objects. This may lead to the potentially confusing situation where the fibre product exists but is not described as below. To avoid this we define the fibre product directly as follows.

Definition 14.7.1. Let $\mathcal{C}$ be a category. Let $U, V, W$ be simplicial objects of $\mathcal{C}$. Let $a : V \to U$, $b : W \to U$ be morphisms. Assume the fibre products $V_ n \times _{U_ n} W_ n$ exist in $\mathcal{C}$. The fibre product of $V$ and $W$ over $U$ is the simplicial object $V \times _ U W$ defined as follows:

  1. $(V \times _ U W)_ n = V_ n \times _{U_ n} W_ n$,

  2. $d^ n_ i = (d^ n_ i, d^ n_ i)$, and

  3. $s^ n_ i = (s^ n_ i, s^ n_ i)$.

In other words, $V \times _ U W$ is the fibre product of the presheaves $V$ and $W$ over the presheaf $U$ on $\Delta $.

Lemma 14.7.2. If $U, V, W$ are simplicial objects in the category $\mathcal{C}$, and if $a : V \to U$, $b : W \to U$ are morphisms and if $V \times _ U W$ exists, then we have

\[ \mathop{\mathrm{Mor}}\nolimits (T, V \times _ U W) = \mathop{\mathrm{Mor}}\nolimits (T, V) \times _{\mathop{\mathrm{Mor}}\nolimits (T, U)} \mathop{\mathrm{Mor}}\nolimits (T, W) \]

for any fourth simplicial object $T$ of $\mathcal{C}$.

Proof. Omitted. $\square$

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