  We show that if $K$ is a nilpotent finite complex, then $\Omega K$ can be
built from spheres using fibrations and homotopy (inverse) limits. This is
applied to show that if ${\mathrm {map}}_*(X,S^n)$ is weakly contractible for
all $n$, then ${\mathrm {map}}_*(X,K)$ is weakly contractible for any nilpotent
finite complex $K$.
