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Doubled topological phases introduced by Kitaev, Levin and Wen supported ontwo dimensional lattices are Hamiltonian versions of three dimensionaltopological quantum field theories described by the Turaev-Viro state summodels. We introduce the latter with an emphasis on obtaining them fromtheories in the continuum. Equivalence of the previous models in the groundstate are shown in case of the honeycomb lattice and the gauge group being afinite group by means of the well-known duality transformation between thegroup algebra and the spin network basis of lattice gauge theory. An analysisof the ribbon operators describing excitations in both types of models and thethree dimensional geometrical interpretation are given.

Microscopic description of 2d topological phases, duality and 3d state sums