FQFT and cohomology
The idea is that a T-fold is a target space for a string sigma-model that is only locally a Riemannian manifold but globally a more general kind of geometry. In the literature sometimes the term non-geometric backgrounds is used for such “generalized geometric” backgrounds.
The idea was originally introduced in
Further developments are in
Aaron Bergman, Daniel Robbins, Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes (arXiv:0710.5158)