We note the following basic facts, which we use without further comment.
For any \(n \geq -2\), a space \(X\) is \(n\)-connected (resp. \(n\)-truncated) if and only if the map \(X \rightarrow{\sf pt}\) is such.
For any \(n \geq -2\), a space is both \(n\)-connected and \(n\)-truncated if and only if it is contractible, and hence a map is both \(n\)-connected and \(n\)-truncated if and only if it is an equivalence.
For any \(n \geq -2\), we have the implications \[\text{$n$-connected} \Longleftarrow \text{$(n+1)$-connected} \qquad \text{and} \qquad \text{$n$-truncated} \Longrightarrow \text{$(n+1)$-truncated}\] for spaces and hence also for maps of spaces.
For any \(n \geq -2\), both \(n\)-connected and \(n\)-truncated maps are stable under base change.
By the long exact sequence in homotopy groups, for any \(n \geq -1\), a map \(X \xrightarrow{f} Y\) of spaces is
\(n\)-connected if and only if for every \(x \in X\) the map \(\pi_i(X,x) \xrightarrow{\pi_i(f)} \pi_i(Y,f(x))\) is
an isomorphism for all \(0 \leq i < n+1\) and
surjective for \(i = n+1\),
and
\(n\)-truncated if and only if for every \(x \in X\) the map \(\pi_i(X,x) \xrightarrow{\pi_i(f)} \pi_i(Y,f(x))\) is
an isomorphism for all \(i > n+1\) and
injective for \(i = n+1\).