Notes on simplicial BF theory

In this work we discuss a construction of “simplicial BF theory” (a field theory with finite-dimensional space of fields, associated to a triangulated manifold), that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of the simplicial program, i.e., the program of constructing discrete topological field theories. We also discuss the relation of these constructions to homotopy algebra.