A pair of homotopy-theoretic version of TQFT's induced by a Brown functor [article]

Minkyu Kim
2021 arXiv   pre-print
The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT's starting from a 𝖧𝗈𝗉𝖿^𝖻𝖼_k-valued Brown functor where 𝖧𝗈𝗉𝖿^𝖻𝖼_k is the category of bicommutative Hopf algebras over a
more » ... ld k : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT's. In another application, we reproduce the Dijkgraaf-Witten TQFT and the Turaev-Viro TQFT from an ordinary 𝖧𝗈𝗉𝖿^𝖻𝖼_k-valued homology theory.
arXiv:2006.10438v3 fatcat:ydchgacdlbc7lh2uioml3tyuce