14.15 Hom from cosimplicial sets into simplicial objects
Let $\mathcal{C}$ be a category. Let $U$ be a cosimplicial object of $\mathcal{C}$, and let $V$ be a simplicial object of $\mathcal{C}$. Then we get a simplicial set $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)$ as follows:
we set $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ n = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U_ n, V_ n)$, and
for $\varphi : [m] \to [n]$ we take the map $\mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ n \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(U, V)_ m$ given by $f \mapsto V(\varphi ) \circ f \circ U(\varphi )$.
This is our motivation for the following definition.
Definition 14.15.1. Let $\mathcal{C}$ be a category with finite products. Let $V$ be a simplicial object of $\mathcal{C}$. Let $U$ be a cosimplicial set such that each $U_ n$ is finite nonempty. We define $\mathop{\mathrm{Hom}}\nolimits (U, V)$ to be the simplicial object of $\mathcal{C}$ defined as follows:
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
\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, \mathop{\mathrm{Hom}}\nolimits (U, V)_ n) = \text{Map}(U_ n, \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, V_ n)) \]
and
for $\varphi : [m] \to [n]$ we take the map $\mathop{\mathrm{Hom}}\nolimits (U, V)_ n \to \mathop{\mathrm{Hom}}\nolimits (U, V)_ m$ 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.
We spell out the construction above in a special case. Let $X$ be an object of a category $\mathcal{C}$. Assume that self products $X \times \ldots \times X$ exist. Let $k$ be an integer. Consider the simplicial object $U$ with terms
\[ U_ n = \prod \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits ([k], [n])} X \]
and maps given $\varphi : [m] \to [n]$
\[ U(\varphi ) : \prod \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits ([k], [n])} X \longrightarrow \prod \nolimits _{\alpha ' \in \mathop{\mathrm{Mor}}\nolimits ([k], [m])} X, \quad (f_{\alpha })_{\alpha } \longmapsto (f_{\varphi \circ \alpha '})_{\alpha '} \]
In terms of “coordinates”, the element $(x_\alpha )_\alpha $ is mapped to the element $(x_{\varphi \circ \alpha '})_{\alpha '}$. We claim this object is equal to $\mathop{\mathrm{Hom}}\nolimits (C[k], X)$ where we think of $X$ as the constant simplicial object $X$ and where $C[k]$ is the cosimplicial set from Example 14.5.6.
Lemma 14.15.2. With $X$, $k$ and $U$ as above.
For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection
\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(V, U) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V_ k, X). \]
wich maps $\gamma $ to the morphism $\gamma _ k$ composed with the projection onto the factor corresponding to $\text{id}_{[k]}$.
Similarly, if $W$ is an $k$-truncated simplicial object of $\mathcal{C}$, then we have
\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ k(\mathcal{C})}(W, \text{sk}_ k U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W_ k, X). \]
The object $U$ constructed above is an incarnation of $\mathop{\mathrm{Hom}}\nolimits (C[k], X)$ where $C[k]$ is the cosimplicial set from Example 14.5.6.
Proof.
We first prove (1). Suppose that $\gamma : V \to U$ is a morphism. This is given by a family of morphisms $\gamma _{\alpha } : V_ n \to X$ for $\alpha : [k] \to [n]$. The morphisms have to satisfy the rules that for all $\varphi : [m] \to [n]$ the diagrams
\[ \xymatrix{ X \ar[d]^{\text{id}_ X} & V_ n \ar[d]^{V(\varphi )} \ar[l]^{\gamma _{\varphi \circ \alpha '}} \\ X & V_ m \ar[l]_{\gamma _{\alpha '}} } \]
commute for all $\alpha ' : [k] \to [m]$. Taking $\alpha ' = \text{id}_{[k]}$, we see that for any $\varphi : [k] \to [n]$ we have $\gamma _\varphi = \gamma _{\text{id}_{[k]}} \circ V(\varphi )$. Thus the morphism $\gamma $ is determined by the component of $\gamma _ k$ corresponding to $\text{id}_{[k]}$. Conversely, given such a morphism $f : V_ k \to X$ we easily construct a morphism $\gamma $ by putting $\gamma _\alpha = f \circ V(\alpha )$.
The truncated case is similar, and left to the reader.
Part (3) is immediate from the construction of $U$ and the fact that $C[k]_ n = \mathop{\mathrm{Mor}}\nolimits ([k], [n])$ which are the index sets used in the construction of $U_ n$.
$\square$
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