Quantum geometry of algebra factorisations and coalgebra bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\text{C})=\text{C}{\mathbb{Z}}_2\cdot \text{C}{\mathbb{Z}}_2$. 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 $S_{q,s}^2$.