  We show that the space of chains of smooth maps from spheres into a fixed
compact oriented manifold has a natural structure of a transversal $d$-algebra.
We construct a structure of transversal 1-category on the space of chains of
maps from a suspension space $S(Y)$, with certain boundary restrictions, into a
fixed compact oriented manifold. We define homological quantum field theories
\HL and construct several examples of such structures. Our definition is based
on the notions of string topology of Chas and Sullivan, and homotopy quantum
field theories of Turaev.
