Proposition 7.6.2.10. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $e: K \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ be a morphism of simplicial sets. Let $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ denote the homotopy coherent nerve of $\operatorname{\mathcal{C}}$, and let $\theta _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$ denote the comparison map of Remark 4.6.8.6. Then:
- $(1)$
The morphism $\theta _{X,Y} \circ e$ exhibits $X$ as a power of $Y$ by $K$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $W \in \operatorname{\mathcal{C}}$, composition with $e$ induces a homotopy equivalence of Kan complexes
\[ c_{W}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W, X)_{\bullet } \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y)_{\bullet } ). \]
- $(2)$
The morphism $\theta _{X,Y} \circ e$ exhibits $Y$ as a tensor product of $X$ by $K$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, precomposition with $e$ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Z)_{\bullet } \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } ). \]
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Fix an object $W \in \operatorname{\mathcal{C}}$, so that the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,X) \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y) \]
of Construction 4.6.9.9 determines a morphism of Kan complexes $c'_{W}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( W,X) \rightarrow \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y)_{\bullet } )$ (which is well-defined up to homotopy). To prove Proposition 7.6.2.10, it will suffice to show that $c'_{W}$ is a homotopy equivalence if and only if $c_{W}$ is a homotopy equivalence. Proposition 4.6.9.19 guarantees that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X)_{\bullet } \ar [r]^-{ c_{W} } \ar [d]^-{ \theta _{W,X} } & \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W, Y)_{\bullet } ) \ar [d]^-{ \theta _{ W, Y} \circ } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( W,X) \ar [r]^-{ c'_{W} } & \operatorname{Fun}(K, \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(W,Y)_{\bullet } } \]
commutes up to homotopy. We conclude by observing that the horizontal maps are homotopy equivalences, by virtue of Theorem 4.6.8.5 (and Remark 4.6.8.6).
$\square$