A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor

The purpose of this paper is to study some obstruction classes induced by a construction homotopy-theoretic version projective TQFT (projective HTQFT for short). A given symmetric monoidal functor whose domain the cospan category pointed finite CW-spaces instead cobordism category. We construct pair HTQFT’s starting from [Formula: see text]-valued Brown where text] bicommutative Hopf algebras over field : cospanical path-integral and spanical functor. They induce an analogue second cohomology class associated with representations. In paper, we derive formulae those classes. apply prove that dimension reduction path-integrals are lifted HTQFT’s. another application, reproduce Dijkgraaf–Witten Turaev–Viro ordinary homology theory.