Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 1.1.4.5. Let $\operatorname{\mathcal{C}}$ be a category and let $C$ be an object of $\operatorname{\mathcal{C}}$. For any simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$, the canonical map

\[ \operatorname{Hom}_{\operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{\mathcal{C}})}( \underline{C}_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \underline{C}_0, X_0) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X_0 ) \]

is a bijection.

Proof. Let $f: C \rightarrow X_0$ be a morphism in $\operatorname{\mathcal{C}}$; we wish to show that $f$ can be promoted uniquely to a map of simplicial objects $f_{\bullet }: \underline{C}_{\bullet } \rightarrow X_{\bullet }$. The uniqueness of $f_{\bullet }$ is clear. For existence, we define $f_{\bullet }$ to be the natural transformation whose value on an object $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ is given by the composite map

\[ \underline{C}_{n} = C \xrightarrow {f} X_0 \xrightarrow { X_{\alpha (n)} } X_{n}, \]

where $\alpha (n)$ denotes the unique morphism in $\operatorname{{\bf \Delta }}$ from $[n]$ to $[0]$. To prove the naturality of $f_{\bullet }$, we observe that for any nondecreasing map $\beta : [m] \rightarrow [n]$ we have a commutative diagram

\[ \xymatrix { \underline{C}_ n \ar@ {=}[r] \ar [d]^{ \underline{C}_{\beta } } & C \ar@ {=}[d] \ar [r]^{f} & X_0 \ar [r]^-{ X_{\alpha (n)} } \ar@ {=}[d] & X_{n} \ar [d]^{ X_{\beta } } \\ \underline{C}_ m \ar@ {=}[r] & C \ar [r]^{f} & X_0 \ar [r]^-{ X_{\alpha (m)} } & X_ m, } \]

where the commutativity of the square on the right follows from the observation that $\alpha (m)$ is equal to the composition $[m] \xrightarrow {\beta } [n] \xrightarrow { \alpha (n) } [0]$. $\square$