Definition 14.10.1. Let $\mathcal{C}$ be a category. Let $U, V, W$ be cosimplicial objects of $\mathcal{C}$. Let $a : V \to U$ and $b : W \to U$ 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 cosimplicial object $V \times _ U W$ defined as follows:

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

for any $\varphi : [n] \to [m]$ the map $(V \times _ U W)(\varphi ) : V_ n \times _{U_ n} W_ n \to V_ m \times _{U_ m} W_ m$ is the product $V(\varphi ) \times _{U(\varphi )} W(\varphi )$.

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