Kerodon

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

Remark 4.6.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. Then the pinch inclusion maps

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

are bijective on vertices: vertices of each simplicial set can be identified with morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remarks 4.6.1.2 and 4.6.5.2). However, they are generally not bijective on edges. Note that edges of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with diagrams

\[ \xymatrix@R =30pt@C=30pt{ X \ar [rrrr]^{f} \ar [dddd]^{\operatorname{id}_{X}} \ar [ddddrrrr]^{g} & & & & Y \ar [dddd]^{\operatorname{id}_{Y}} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ X \ar [rrrr]^{f'} & & & & Y } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. Such a diagram belongs to the image of the left-pinch inclusion map $\iota ^{\mathrm{L}}_{X,Y}$ if and only $\tau = s_0( g )$ (so that the simplex $\tau $ is degenerate, $f' = g$, and the entire diagram is determined by $\sigma $). Similarly, the diagram belongs to the image of the right-pinch inclusion map $\iota ^{\mathrm{R}}_{X,Y}$ if and only if $\sigma = s_1(g)$ (so that the simplex $\sigma $ is degenerate, $f = g$, and the entire diagram is determined by $\tau $).