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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 HTQFTs starting from a $mathsf{Hopf}^mathsf{bc}_k$-valued Brown functor where $mathsf{Hopf}^mathsf{bc}_k$ is the category of bicommutative Hopf algebras over a field $k$ : 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 HTQFTs. In another application, we reproduce the Dijkgraaf-Witten TQFT and the Turaev-Viro TQFT from an ordinary $mathsf{Hopf}^mathsf{bc}_k$-valued homology theory.
Let $mathcal{A}$ be a small abelian category. The purpose of this paper is to introduce and study a category $overline{mathcal{A}}$ which implicitly appears in construction of some TQFTs where $overline{mathcal{A}}$ is determined by $mathcal{A}$. If
We classify a number of symmetry protected phases using Freed-Hopkins homotopy theoretic classification. Along the way we compute the low-dimensional homotopy groups of a number of novel cobordism spectra.
There exists a canonical functor from the category of fibrant objects of a model category modulo cylinder homotopy to its homotopy category. We show that this functor is faithful under certain conditions, but not in general.
Let $M$ be a topological monoid with homotopy group completion $Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $pi_k (Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of n
In this paper, Wielandts inequality for classical channels is extended to quantum channels. That is, an upper bound to the number of times a channel must be applied, so that it maps any density operator to one with full rank, is found. Using this bou