Turaev-Viro invariants as an extended TQFT

In this paper we show how one can extend Turaev-Viro invariants, defined for an arbitrary spherical fusion category $C$, to 3-manifolds with corners. We demonstrate that this gives an extended TQFT which conjecturally coincides with the Reshetikhin-Turaev TQFT corresponding to the Drinfeld center $Z(C)$. In the present paper we give a partial proof of this statement.
Comment: 38 pages, many figures. New in v.3: significantly rewritten exposition, providing details of some proofs which were only outlined in the previous version