The Stacks project

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

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

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