Dimension if and only if connected objects have sections

For the `only if'-direction, suppose \(T\) has dimension \(\leq d\), and let \(X\) be a \((d-1)\)-connected object. Then the unique map \(X \to *\) is \((d-1)\)-connected. By the dimension condition, the induced map \(\Gamma_*(X) \to \Gamma_*(*) = *\) is \((-1)\)-connected. A map to the terminal object being \((-1)\)-connected means that its domain is nonempty, so \(X\) has a global section.For the `if'-direction, suppose every \((d-1)\)-connected object of \(T\) has a global section. We prove by induction on \(n \geq d-1\) that \(\Gamma_*\) sends \(n\)-connected maps to \((n-d)\)-connected maps.For the initial case, let \(f\colon X \to Y\) be \((d-1)\)-connected. An effective epimorphism of animae is characterized by having nonempty fibers over every point. Given \(y\colon * \to Y\), the fiber of \(\Gamma_*(f)\) over \(y\) is where \(X_y := X \times_Y *\). Since \(X_y\) is \((d-1)\)-connected, it has a global section by assumption. Hence the displayed fiber is nonempty, so \(\Gamma_*(f)\) is an effective epimorphism, i.e. it is \((-1)\)-connected.Now let \(n \geq d\) and let \(f\) be \(n\)-connected. It is in particular \((d-1)\)-connected, so \(\Gamma_*(f)\) is an effective epimorphism. Since \(\Gamma_*\) preserves limits, the diagonal of \(\Gamma_*(f)\) is \(\Gamma_*(\Delta_f)\). The map \(\Delta_f\) is \((n-1)\)-connected, so the induction hypothesis shows that its image is \((n-d-1)\)-connected. The characterization of connected maps by their diagonals in Theorem 3.22 now shows that \(\Gamma_*(f)\) is \((n-d)\)-connected.