Photons do not in general interact. However, in a cavity QED system, a qubit can mediate an effective interaction between photons. In coupled arrays of such cQED elements, photons can hop between neighboring cavities and experience on-site interactions in any one cavity. This interaction is similar to the interactions that occur in atomic lattices. By using cavity arrays, qubits, and other superconducting elements, we can simulate various lattices. This extends to the simulation of interesting geometries, like hyperbolic space, that cannot classically exist in classical Euclidean space.