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*}

14.14 Hom from simplicial sets into cosimplicial objects

Let $\mathcal{C}$ be a category. Let $U$ be a simplicial object of $\mathcal{C}$, and let $V$ be a cosimplicial object of $\mathcal{C}$. Then we get a cosimplicial set $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)$ as follows:

  1. we set $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ n = \mathop{Mor}\nolimits _\mathcal {C}(U_ n, V_ n)$, and

  2. for $\varphi : [m] \to [n]$ we take the map $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ m \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ n$ given by $f \mapsto V(\varphi ) \circ f \circ U(\varphi )$.

This is our motivation for the following definition.

Definition 14.14.1. Let $\mathcal{C}$ be a category with finite products. Let $V$ be a cosimplicial object of $\mathcal{C}$. Let $U$ be a simplicial set such that each $U_ n$ is finite nonempty. We define $\mathop{\mathrm{Hom}}\nolimits (U, V)$ to be the cosimplicial object of $\mathcal{C}$ defined as follows:

  1. we set $\mathop{\mathrm{Hom}}\nolimits (U, V)_ n = \prod _{u \in U_ n} V_ n$, in other words the unique object of $\mathcal{C}$ such that its $X$-valued points satisfy

    \[ \mathop{Mor}\nolimits _\mathcal {C}(X, \mathop{\mathrm{Hom}}\nolimits (U, V)_ n) = \text{Map}(U_ n, \mathop{Mor}\nolimits _\mathcal {C}(X, V_ n)) \]

    and

  2. for $\varphi : [m] \to [n]$ we take the map $\mathop{\mathrm{Hom}}\nolimits (U, V)_ m \to \mathop{\mathrm{Hom}}\nolimits (U, V)_ n$ given by $f \mapsto V(\varphi ) \circ f \circ U(\varphi )$ on $X$-valued points as above.

We leave it to the reader to spell out the definition in terms of maps between products. We also point out that the construction is functorial in both $U$ (contravariantly) and $V$ (covariantly), exactly as in Lemma 14.13.3 in the case of products of simplicial sets with simplicial objects.


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