Proposition 1.2.3.18 (00H5)

Proposition 1.2.3.18. Let $S$ be a simplicial set. The following conditions are equivalent:

$(1)$: The geometric realization $| S |$ is a path-connected topological space.
$(2)$: The geometric realization $| S |$ is a connected topological space.
$(3)$: The simplicial set $S$ is connected, in the sense of Definition 1.2.1.6.

Proof. The implication $(1) \Rightarrow (2)$ holds for any topological space. To prove that $(2) \Rightarrow (3)$, we observe that any decomposition $S \simeq S' \coprod S''$ into disjoint nonempty simplicial subsets determines a homeomorphism $| S | \simeq | S' | \coprod | S'' |$. We will complete the proof by showing that $(3) \Rightarrow (1)$. Let $\operatorname{{\bf \Delta }}_{S}$ denote the category of simplices of $S$ (Construction 1.1.3.9). We then have a commutative diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \varinjlim _{([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } | \Delta ^ n | \ar [r]^-{\sim } \ar [d] & | S | \ar [d] \\ \varinjlim _{ ([n], \sigma ) \in \operatorname{{\bf \Delta }}_{S} } \pi _0( | \Delta ^ n | ) \ar [r] & \pi _0( | S | ), } \]

where the upper horizontal map is bijective (Remark 1.2.3.16) and the right vertical map is surjective. It follows that the lower horizontal map is also surjective. Since each of the topological spaces $| \Delta ^ n |$ is path connected, the colimit in the lower left can be identified with the set $\pi _0( S )$ (Remark 1.2.3.17). If $S$ is connected, the set $\pi _0(S)$ consists of a single element, so that $\pi _0( | S | )$ is also a singleton. $\square$