For any \(n \geq 0\), a space \(X \in \mathcal S\) is called
\(n\)-connected if \(X\) is connected and if \(\pi_i(X,x) = 0\) for all \(i \leq n\) and all \(x \in X\) and
\(n\)-truncated if \(\pi_i(X,x) = 0\) for all \(i > n\) and all \(x \in X\).
We extend this to the case that \(n=-1\) by declaring that \(X\) is
\((-1)\)-connected if it is nonempty and
\((-1)\)-truncated if it is either empty or contractible,
and to the case that \(n=-2\) by declaring that \(X\) is
always \((-2)\)-connected and
\((-2)\)-truncated if it is contractible.
For any \(n \geq -2\), we declare that a map of spaces is \(n\)-connected26 (resp. \(n\)-truncated) if its fibers are all such. By [Lur09, Ex. 5.2.8.16] the classes of (\(n\)-connected, \(n\)-truncated) maps form a factorization system of small generation on \(\mathcal S\), generated by the single morphism \(S^{n+1} \rightarrow{\sf pt}\). See also example B.1.16.