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*}

14.13 Products with simplicial sets

Let $\mathcal{C}$ be a category. Let $U$ be a simplicial set. Let $V$ be a simplicial object of $\mathcal{C}$. We can consider the covariant functor which associates to a simplicial object $W$ of $\mathcal{C}$ the set

14.13.0.1
\begin{equation} \label{simplicial-equation-functor-product-with-simplicial-set} \left\{ (f_{n, u} : V_ n \to W_ n)_{n \geq 0, u \in U_ n} \text{ such that } \begin{matrix} \forall \varphi : [m] \to [n] \\ f_{m, U(\varphi )(u)} \circ V(\varphi ) = W(\varphi ) \circ f_{n, u} \end{matrix} \right\} \end{equation}

If this functor is of the form $\mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(Q, -)$ then we can think of $Q$ as the product of $U$ with $V$. Instead of formalizing this in this way we just directly define the product as follows.

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.

Lemma 14.13.2. 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. The functor $W \mapsto \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(U \times V, W)$ is canonically isomorphic to the functor which maps $W$ to the set in Equation (14.13.0.1).

Proof. Omitted. $\square$

Lemma 14.13.3. Let $\mathcal{C}$ be a category such that the coproduct of any two objects of $\mathcal{C}$ exists. Let us temporarily denote $\textit{FSSets}$ the category of simplicial sets all of whose components are finite nonempty.

  1. The rule $(U, V) \mapsto U \times V$ defines a functor $\textit{FSSets} \times \text{Simp}(\mathcal{C}) \to \text{Simp}(\mathcal{C})$.

  2. For every $U$, $V$ as above there is a canonical map of simplicial objects

    \[ U \times V \longrightarrow V \]

    defined by taking the identity on each component of $(U \times V)_ n = \coprod _ u V_ n$.

Proof. Omitted. $\square$

We briefly study a special case of the construction above. Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$. Let $k \geq 0$ be an integer. If all coproducts $X \amalg \ldots \amalg X$ exist then according to the definition above the product

\[ X \times \Delta [k] \]

exists, where we think of $X$ as the corresponding constant simplicial object.

Lemma 14.13.4. With $X$ and $k$ as above. For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection

\[ \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(X \times \Delta [k], V) \longrightarrow \mathop{Mor}\nolimits _\mathcal {C}(X, V_ k). \]

which maps $\gamma $ to the restriction of the morphism $\gamma _ k$ to the component corresponding to $\text{id}_{[k]}$. Similarly, for any $n \geq k$, if $W$ is an $n$-truncated simplicial object of $\mathcal{C}$, then we have

\[ \mathop{Mor}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ n(X \times \Delta [k]), W) = \mathop{Mor}\nolimits _\mathcal {C}(X, W_ k). \]

Proof. A morphism $\gamma : X \times \Delta [k] \to V$ is given by a family of morphisms $\gamma _\alpha : X \to V_ n$ where $\alpha : [n] \to [k]$. The morphisms have to satisfy the rules that for all $\varphi : [m] \to [n]$ the diagrams

\[ \xymatrix{ X \ar[r]^{\gamma _\alpha } \ar[d]^{\text{id}_ X} & V_ n \ar[d]^{V(\varphi )} \\ X \ar[r]^{\gamma _{\alpha \circ \varphi }} & V_ m } \]

commute. Taking $\alpha = \text{id}_{[k]}$, we see that for any $\varphi : [m] \to [k]$ we have $\gamma _\varphi = V(\varphi ) \circ \gamma _{\text{id}_{[k]}}$. Thus the morphism $\gamma $ is determined by the value of $\gamma $ on the component corresponding to $\text{id}_{[k]}$. Conversely, given such a morphism $f : X \to V_ k$ we easily construct a morphism $\gamma $ by putting $\gamma _\alpha = V(\alpha ) \circ f$.

The truncated case is similar, and left to the reader. $\square$

A particular example of this is the case $k = 0$. In this case the formula of the lemma just says that

\[ \mathop{Mor}\nolimits _\mathcal {C}(X, V_0) = \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(X, V) \]

where on the right hand side $X$ indicates the constant simplicial object with value $X$. We will use this formula without further mention in the following.


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