Motivated by experiments performed by Landig et al. [Nature (London) 532, 476 (2016)NATUAS0028-083610.1038/nature17409], we consider a two-dimensional Bose gas in an optical lattice, trapped inside a single mode superradiant Fabry-Perot cavity. The cavity mediates infinite-range checkerboard interactions between the atoms, which produces competition between Mott insulator, charge-density wave, superfluid, and supersolid phases. We calculate the phase diagram of this Bose gas in a homogeneous system and in the presence of a harmonic trap. © 2016 American Physical Society.