Notation 2.4 (0KE9) · ScalingStacks

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Notation 2.4. Let $A\subseteq B$ and $D$ be simplicial sets. We define $B\rtimes_{A}D$ by the following pushout diagram

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Let $A\subseteq B$ and $X$ be simplicial sets. Given two maps $f,g\colon B\to X$ such that $f|_{A}=g|_{A}$ we obtain a map $f\cup g\colon B\rtimes_{A}\partial\Delta^{1}\to X$ . A homotopy relative to $A$ (or “rel. $A$ ” for short) is an extension of $f\cup g$ to $B\rtimes_{A}\Delta^{1}$ .
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Given inclusions of simplicial sets $A\subseteq B\subseteq C$ and a simplicial set $X$ , let $\left[B,C;X\right]$ be the set of maps $B\to X$ for which there exists an extension to $C$ . We denote by $\left[A,B,C;X\right]$ the set obtained from $\left[B,C;X\right]$ by identifying maps that are homotopic rel. $A$ .