The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

14.6 Products of simplicial objects

Of course we should define the product of simplicial objects as the product in the category of simplicial 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 product directly as follows.

Definition 14.6.1. Let $\mathcal{C}$ be a category. Let $U$ and $V$ be simplicial objects of $\mathcal{C}$. Assume the products $U_ n \times V_ n$ exist in $\mathcal{C}$. The product of $U$ and $V$ is the simplicial object $U \times V$ defined as follows:

  1. $(U \times V)_ n = U_ n \times V_ n$,

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

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

In other words, $U \times V$ is the product of the presheaves $U$ and $V$ on $\Delta $.

Lemma 14.6.2. If $U$ and $V$ are simplicial objects in the category $\mathcal{C}$, and if $U \times V$ exists, then we have

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

for any third simplicial object $W$ of $\mathcal{C}$.

Proof. Omitted. $\square$


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