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*}

14.16 Internal Hom

Let $\mathcal{C}$ be a category with finite nonempty products. Let $U$, $V$ be simplicial objects $\mathcal{C}$. In some cases the functor

\[ \text{Simp}(\mathcal{C})^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(W \times V, U) \]

is representable. In this case we denote $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (V, U)$ the resulting simplicial object of $\mathcal{C}$, and we say that the internal hom of $V$ into $U$ exists. Moreover, in this case, given $X$ in $\mathcal{C}$, we would have

\begin{align*} \mathop{Mor}\nolimits _\mathcal {C}(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (V, U)_ n) & = \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(X \times \Delta [n], \mathop{\mathcal{H}\! \mathit{om}}\nolimits (V, U)) \\ & = \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(X \times \Delta [n]\times V, U) \\ & = \mathop{Mor}\nolimits _{\text{Simp}(\mathcal{C})}(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\Delta [n] \times V, U)) \\ & = \mathop{Mor}\nolimits _\mathcal {C}(X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\Delta [n] \times V, U)_0) \end{align*}

provided that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\Delta [n] \times V, U)$ exists also. The first and last equalities follow from Lemma 14.13.4.

The lesson we learn from this is that, given $U$ and $V$, if we want to construct the internal hom then we should try to construct the objects

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\Delta [n] \times V, U)_0 \]

because these should be the $n$th term of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (V, U)$. In the next section we study a construction of simplicial objects “$\mathop{\mathrm{Hom}}\nolimits (\Delta [n], U)$”.


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