Remark 2.5 (0KEA) · ScalingStacks

[0KEA]

Remark 2.5. Let $\mathcal{C}$ be an $\infty$ -category, let $A\subseteq B$ be an inclusion of simplicial sets, and consider $f,g\colon B\to\mathcal{C}$ such that $f|_{A}=g|_{A}$ . By the discussion at the beginning of T.2.3.4, a homotopy from $f$ to $g$ rel. $A$ is the same as an equivalence from $f$ to $g$ as objects of the $\infty$ -category $\mathcal{D}$ that is given as a pullback

Therefore, the existence of a homotopy rel. $A$ is an equivalence relation. We note that the above diagram is also a homotopy pullback in the Joyal model structure as the right vertical map is a categorical fibration and all objects are fibrant.