  We compute the $L_\infty$-theoretic dimensional reduction of the
F1/D$p$-brane super $L_\infty$-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 $L_\infty$-algebras are
naturally related by an $L_\infty$-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 $K^0$ and $K^1$,
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
$L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by
the "T-duality Lie 2-algebra". We find that the induced $L_\infty$-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 observe that
this satisfies expected properties of a local model space for F-theory elliptic
fibrations.
