Corollary 1.2.1.12 (00GH)

Corollary 1.2.1.12. Let $S_{}$ be a simplicial set. The following conditions are equivalent:

$(a)$

The simplicial set $S_{}$ is connected.

$(b)$

For every set $I$, the canonical map

\[ I \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^0, \underline{I}_{} ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{}, \underline{I}_{} ) \]

is bijective.

Proof. The implication $(a) \Rightarrow (b)$ follows from Proposition 1.2.1.11 and Example 1.2.1.10. Conversely, suppose that $(b)$ is satisfied. Applying $(b)$ in the case $I = \emptyset $, we conclude that there are no maps from $S_{}$ to the empty simplicial set, so that $S_{}$ is nonempty. If $S_{}$ is a disjoint union of simplicial subsets $S'_{}, S''_{} \subseteq S_{}$, then we obtain a map of simplicial sets

\[ S_{} \simeq S'_{} \coprod S''_{} \rightarrow \Delta ^0 \coprod \Delta ^0 \]

and assumption $(b)$ guarantees that this map factors through one of the summands on the right hand side; it follows that either $S'_{}$ or $S''_{}$ is empty. $\square$