abstract duality: opposite category,
A T-fold (Hull 04) is supposed to be a kind of space that locally looks like a Riemannian manifold equipped with a B-field, but which is is glued together from these not just by diffeomorphisms but also by T-duality transformations along some torus fibers.
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.
It is expected that T-folds should have a description in terms of spaces that locally are fiber products of one torus fiber bundle with its T-dual. (One proposed formalization is that these are the total spaces of principal 2-bundles for the T-duality 2-group).
The idea was originally introduced in
The relation to double field theory goes back to
Further developments are in
Aaron Bergman, Daniel Robbins, Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes (arXiv:0710.5158)