Kerodon

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

Proposition 5.7.6.17. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let

\[ \mathscr {F}: \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}) = \operatorname{\mathcal{S}}. \]

denote the homotopy coherent nerve of the simplicial functor $Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. Then the identity morphism $\operatorname{id}_{X} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)_{\bullet } = \mathscr {F}(X)$ exhibits the functor $\mathscr {F}$ as corepresented by $X$, in the sense of Definition 5.7.6.1.

Proof. Fix an object $Y \in \operatorname{\mathcal{C}}$. We then have a commutative diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [r]^-{U} \ar [d]^{\theta }_{\sim } & \operatorname{Hom}_{ \operatorname{Kan}}( \mathscr {F}(X), \mathscr {F}(Y) )_{\bullet } \ar [d]^{\theta '}_{\sim } \ar [r]^-{\operatorname{ev}} & \mathscr {F}(Y) \\ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \ar [r]^-{V} & \operatorname{Hom}_{ \operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {F}(Y) ), & } \]

where the vertical maps are supplied by Construction 4.6.7.3 (applied in the simplicial categories $\operatorname{\mathcal{C}}$ and $\operatorname{Kan}$, respectively) and $\operatorname{ev}$ is given by evaluation at the vertex $\operatorname{id}_{X} \in \mathscr {F}(X)$. Let $\theta '^{-1}$ denote a homotopy inverse to $\theta '$ (which exists by virtue of Theorem 4.6.7.5). Proposition 5.7.6.17 asserts that the composition $\operatorname{ev}\circ \theta '^{-1} \circ V$ is a homotopy equivalence. Since $\theta $ is also a homotopy equivalence (Theorem 4.6.7.5), this is equivalent to the assertion that $\operatorname{ev}\circ U$ is a homotopy equivalence. This is clear: the composition $\operatorname{ev}\circ U$ is the identity map from the Kan complex $\mathscr {F}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to itself. $\square$