Kerodon

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

Corollary 4.6.4.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$. Let $A$ be a simplicial subset of $B$ satisfying $f_0|_{A} = f_1|_{A}$, and let

\[ (B \coprod _{A} B) \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]

be a categorical mapping cylinder of $B$ relative to $A$. The following conditions are equivalent:

$(a)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$.

$(b)$

There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.

In particular, condition $(b)$ does not depend on the choice of categorical mapping cylinder.

Proof. Apply Corollary 4.6.4.10 in the special case $S = \Delta ^{0}$. $\square$