Kerodon

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

Proposition 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinch inclusion morphisms

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \xrightarrow { \iota ^{\mathrm{L}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xleftarrow { \iota ^{\mathrm{R}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]

are homotopy equivalences of Kan complexes.

Proof. We will prove that the left-pinch inclusion morphism $\iota ^{\mathrm{L}}_{X,Y}$ is a homotopy equivalence; the proof for the right-pinch inclusion morphism $\iota ^{\mathrm{R}}_{X,Y}$ is similar. Note that we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r]^-{ \iota ^{\mathrm{L}}_{X} } \ar [d] & (X \downarrow \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, } \]

where the horizontal maps are equivalences of $\infty $-categories (Theorem 4.6.5.14) and the vertical maps are left fibrations (Propositions 4.3.6.1 and 4.6.5.10), hence isofibrations (Example 4.4.1.10). Applying Corollary 4.5.4.4, we deduce that the induced map of fibers

\[ \iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) = (\operatorname{\mathcal{C}}_{X/} ) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow (X \downarrow \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is an equivalence of $\infty $-categories, hence a homotopy equivalence of Kan complexes (Remark 4.5.1.4). $\square$