Connected objects and morphisms

Let \(T\) be a topos, and let \(n \geq -2\).

• We say that \(X \in T\) is \(n\)-connected if \(\tau_n X = *\).

• We say \(f \colon X \to Y\) is \(n\)-connected if it is \(n\)-connected in \(T_{/Y}\).

We denote by \(T^{\geq n+1}\) the full subcategory of \(T\) spanned by the \(n\)-connected objects.