  In this work we discuss the simplicial program for topological field theories
for the case of non-abelian BF theory. Discrete BF theory with
finite-dimensional space of fields is constructed for a triangulated manifold
(or for a manifold equipped with cubical cell decomposition), that is in a
sense equivalent to the topological BF theory on manifold. This discrete
version allows one to calculate interesting quantities from the BF theory, like
the effective action on cohomology, in terms of finite-dimensional integrals
instead of functional integrals, as demonstrated in a series of explicit
examples. We also discuss the interpretation of discrete BF action as the
generating function for $qL_\infty$ structure (certain "one-loop version" of
ordinary $L_\infty$ algebra) on the cell cochains of triangulation, related to
the de Rham algebra of the underlying manifold by homotopy transfer procedure.
This work is a refinement of older text hep-th/0610326.
