Example 6.72.
The presheaf category \(\PSh(\N)\) on the poset of natural numbers has dimension \(\leq 1\). Indeed, a \(0\)-connected presheaf is a tower
\[\cdots \longrightarrow X_2 \longrightarrow X_1 \longrightarrow X_0\]
of connected animae. Its limit is nonempty: after replacing the tower by a tower of Kan fibrations, one may choose a point of \(X_0\) and lift it successively through the tower. Thus every \(0\)-connected object has a global section, and Lemma 6.68 applies.