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 $\Delta$.

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