To obtain examples, the following explicit alternative descriptions of \(n\)-connectedness and \(n\)-truncatedness for low values of \(n\) are useful.
A space is \(0\)-connected if and only if it is connected (and in particular nonempty), and it is \(1\)-connected if and only if it is simply connected (and in particular connected).
A map of spaces is always \((-2)\)-connected, and it is \((-1)\)-connected if and only if it is surjective.
A space is \(n\)-truncated if and only if it is an \(n\)-type, e.g. it is \(0\)-truncated if and only if it is discrete.
A map of spaces is \((-2)\)-truncated if and only if it is an equivalence, it is \((-1)\)-truncated if and only if it is a monomorphism, and it is \(0\)-truncated if and only if it is a covering map (in the classical sense).