On Higher Topological T-duality Functors

We use String Field Theory (SFT) to construct a higher analogue of Bunke-Schick's functor $P: \mathbf{Top}^{op} \to \mathbf{Set}$ \cite{BunkeS1} by geometrizing $P.$ We use the projection of SFT onto its massless modes \cite{SFTDiffeo} to construct the category $\C$ whose objects are pairs (which we identify with SFT backgrounds) and whose maps are morphisms of pairs (which are gauge transformations). Using $\C$ and categorical equivalence, for any $CW-$complex $X$ we define the moduli space $G(X)$ of SFT backgrounds which are pairs over $X$ up to gauge equivalence. We use the homotopy theory of the moduli space $G(X)$ to define functors on the category of $CW-$complexes $P_k:\mathbf{CW}^{op} \to \mathbf{Grpd}$ such that $P_0 \simeq P,$ $P_1$ is nontrivial and $P_k(X)$ are always trivial for $k \geq 2.$ Arrows in $P_1(X)$ are shown to be isotopy classes of maps in the mapping class group of $X$ acting on (isomorphism classes of) pairs over $X.$ We discuss applications to Topological T-duality for triples and to modelling doubled geometries and T-folds \cite{HullT}.