  A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs
is extended by new generators called fusing, braiding, twist and switch with
relations which can be called Moore--Seiberg relations. A functor to N is
constructed from the category Surf of oriented surfaces with labeled boundary
and their homeomorphisms. Given an (eventually non-semisimple) k-linear abelian
ribbon braided category C with some finiteness conditions we construct a
functor from a central extension of N with the set of labels ObC to k-vector
spaces. Composing the functors we get a modular functor from a central
extension of Surf to k-vector spaces. This is a mathematical paper which
explains how to get proofs for its hep-th companion paper, which should be read
first. Complete proofs are not given here. (Talk at Second Gauss Simposium,
Munich, August 1993.)
