  We develop the noncommutative geometry (bundles, connections etc.) associated
to algebras that factorise into two subalgebras. An example is the
factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend
the coalgebra version of theory introduced previously, to include frame
resolutions and corresponding covariant derivatives and torsions. As an
example, we construct $q$-monopoles on all the Podle\'s quantum spheres
$S^2_{q,s}$.
