T-Duality from super Lie $n$-algebra cocycles for super $p$-branes

We compute the L-infinity-theoretic double dimensional reduction of the F1/Dp-brane super L-infinity-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient L-infinity-algebras are naturally related by an L-infinity-isomorphism which we find to act on the super p-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between K-0-cocycles in type IIA and K-1-cocycles in type IIB, rationally. In particular this is a derivation of the Buscher rules for RR-fields (Hori's formula) from first principles. Moreover, we show that these L-infinity-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra" . We find that the induced L-infinity-extension is a gerby extension of a 9 + (1 + 1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10 + (1 + 1)-dimensional super Lie algebra. Finally we show that this satisfies expected properties of a local model space for F-theory elliptic fibrations.