Lemma 14.17.2 (017J)—The Stacks project

Lemma 14.17.2. Assume the category $\mathcal{C}$ has coproducts of any two objects and countable limits. Let $U$ be a simplicial set, with $U_ n$ finite nonempty for all $n \geq 0$. Let $V$ be a simplicial object of $\mathcal{C}$. Then the functor

\begin{eqnarray*} \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ X & \longmapsto & \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(X \times U, V) \end{eqnarray*}

is representable.

Proof. A morphism from $X \times U$ into $V$ is given by a collection of morphisms $f_ u : X \to V_ n$ with $n \geq 0$ and $u \in U_ n$. And such a collection actually defines a morphism if and only if for all $\varphi : [m] \to [n]$ all the diagrams

\[ \xymatrix{ X \ar[r]^{f_ u} \ar[d]_{\text{id}_ X} & V_ n \ar[d]^{V(\varphi )} \\ X \ar[r]^{f_{U(\varphi )(u)}} & V_ m } \]

commute. Thus it is natural to introduce a category $\mathcal{U}$ and a functor $\mathcal{V} : \mathcal{U}^{opp} \to \mathcal{C}$ as follows:

The set of objects of $\mathcal{U}$ is $\coprod _{n \geq 0} U_ n$,
a morphism from $u' \in U_ m$ to $u \in U_ n$ is a $\varphi : [m] \to [n]$ such that $U(\varphi )(u) = u'$
for $u \in U_ n$ we set $\mathcal{V}(u) = V_ n$, and
for $\varphi : [m] \to [n]$ such that $U(\varphi )(u) = u'$ we set $\mathcal{V}(\varphi ) = V(\varphi ) : V_ n \to V_ m$.

At this point it is clear that our functor is nothing but the functor defining

\[ \mathop{\mathrm{lim}}\nolimits _{\mathcal{U}^{opp}} \mathcal{V} \]

Thus if $\mathcal{C}$ has countable limits then this limit and hence an object representing the functor of the lemma exist. $\square$

The Stacks project

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