  We formulate two-dimensional rational conformal field theory as a natural
generalization of two-dimensional lattice topological field theory. To this end
we lift various structures from complex vector spaces to modular tensor
categories. The central ingredient is a special Frobenius algebra object A in
the modular category that encodes the Moore-Seiberg data of the underlying
chiral CFT. Just like for lattice TFTs, this algebra is itself not an
observable quantity. Rather, Morita equivalent algebras give rise to equivalent
theories. Morita equivalence also allows for a simple understanding of
T-duality.
  We present a construction of correlators, based on a triangulation of the
world sheet, that generalizes the one in lattice TFTs. These correlators are
modular invariant and satisfy factorization rules. The construction works for
arbitrary orientable world sheets, in particular for surfaces with boundary.
Boundary conditions correspond to representations of the algebra A. The
partition functions on the torus and on the annulus provide modular invariants
and NIM-reps of the fusion rules, respectively.
