Definition 14.13.1. Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\mathcal{C}$. Assume that each $U_ n$ is finite nonempty. In this case we define the product $U \times V$ of $U$ and $V$ to be the simplicial object of $\mathcal{C}$ whose $n$th term is the object

$(U \times V)_ n = \coprod \nolimits _{u\in U_ n} V_ n$

with maps for $\varphi : [m] \to [n]$ given by the morphism

$\coprod \nolimits _{u\in U_ n} V_ n \longrightarrow \coprod \nolimits _{u'\in U_ m} V_ m$

which maps the component $V_ n$ corresponding to $u$ to the component $V_ m$ corresponding to $u' = U(\varphi )(u)$ via the morphism $V(\varphi )$. More loosely, if all of the coproducts displayed above exist (without assuming anything about $\mathcal{C}$) we will say that the product $U \times V$ exists.

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