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14.10 Fibre products of cosimplicial objects

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

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:

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

  2. 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 $V(\varphi ) \times _{U(\varphi )} W(\varphi )$.

Lemma 14.10.2. If $U, V, W$ are cosimplicial 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 cosimplicial object $T$ of $\mathcal{C}$.

Proof. Omitted. $\square$

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