Remark 3.5.7.8. Proposition 3.5.7.7 is also true in the case $n = -1$, provided that restate condition $(\ast _ m)$ as follows:
- $(\ast '_ m)$
For every vertex $x \in X$, the set $\pi _{m}(X,x)$ consists of a single element.
Note that $(\ast '_0)$ is equivalent to the assertion that every pair of vertices of $X$ belong to the same connected component: that is, every morphism $\operatorname{\partial \Delta }^1 \rightarrow X$ can be extended to a $1$-simplex of $X$.