Sorry for having referred to the article for details towards the end. I now went and added a little more text to go with that last diagram. Because the reason for its hour glass look is not that there is a bottleneck in the extension process here, but because towards the right the diagram shows cocycles, and to the left it shows the homotopy fibers of these cocycles.
Namely on the right is shown the double dimensional reduction of the F1/Dp-brane cocycles for d=10 type IIA and type IIB, respectively. The triangle on the right expresses that down in 9 these become equivalent. This is T-duality on the cocycle level. Specifically, it turns out that in components this equivalence is the Buscher rule for RR-fields (derived thereby, from first principles). Now by functoriality of homotopy fibers, also the extensions classified by these cocycles become equivalent, and this is expressed by the triangle on the left. These equivalent extensions thus classified turn out to be the doubled superspacetimes with their B-field generalized geometry, and their equivalence is hence T-duality made manifest as a symmetry of the doubled spacetime.
Wish my math was up to the full details these days.
It taps out at RHS's - that's about my limit.
Thanks for the feedback.
Let me amplify that while homotopy theory in general and rational homotopy theory (RHS) in particular provide the full story that I am sketching above, most of the results surveyed above may be seen already via elementary means by considering just "FDA"s as used in supergravity, augmented only by the standard algorithm for computing homotopy (co-)fibers. The lecture notes at ncatlab.org/nlab/print/geometry+of+physics+--+fundamental+super+p-branes are written for an audience with no particular background, are expository, detailed and self-contained. And for something half-way between the terse slides for the above and these full lecture notes, there is also these seminar notes: ncatlab.org/schreiber/print/Super+Lie+n-algebra+of+Super+p-branes.
(Best viewed with Firefox or one of its derivatives, since other browsers will call MathJax to render the formulas, which then takes ages.)
We establish a higher generalization of super L-∞-algebraic T-duality of super WZW-terms for super p-branes. In particular we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d superspacetimes. Finally we observe that this constitutes a duality-isomorphism relating a priori different moduli spaces for C-field configurations in exceptional generalized geometry.