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

Remark Let $B$ be a simplicial set and let $A$ be a simplicial subset. Then the simplicial set $B/A$ can be described more informally as follows: it is obtained from $B$ by collapsing the simplicial subset $A \subseteq B$ to a single vertex $q_0$. Beware that this informal description is a bit misleading when $A = \emptyset $: in this case, the natural map $B \rightarrow B/A$ is not surjective (instead, $B/A$ can be described as the coproduct $B_{+} = B \coprod \{ q_0 \} $, obtained from $B$ by adding a new base point).