  We construct combinatorial analogs of 2d higher topological quantum field
theories. We consider triangulations as vertices of a certain CW complex $\Xi$.
In the "flip theory," cells of $\Xi_\mathrm{flip}$ correspond to polygonal
decompositions obtained by erasing the edges in a triangulation. These theories
assign to a cobordism $\Sigma$ a cochain $Z$ on $\Xi_\mathrm{flip}$ constructed
as a contraction of structure tensors of a cyclic $A_\infty$ algebra $V$
assigned to polygons. The cyclic $A_\infty$ equations imply the closedness
equation $(\delta+Q)Z=0$. In this context we define combinatorial BV operators
and give examples with coefficients in $\mathbb{Z}_2$. In the "secondary
polytope theory," $\Xi_\mathrm{sp}$ is the secondary polytope (due to
Gelfand-Kapranov-Zelevinsky) and the cyclic $A_\infty$ algebra has to be
replaced by an appropriate refinement that we call an $\widehat{A}_\infty$
algebra. We conjecture the existence of a good Pachner CW complex $\Xi$ for any
cobordism, whose local combinatorics is descibed by secondary polytopes and the
homotopy type is that of Zwiebach's moduli space of complex structures.
Depending on this conjecture, one has an "ideal model" of combinatorial 2d
HTQFT determined by a local $\widehat{A}_\infty$ algebra.
