  Kaluza-Klein Theory states that a metric on the total space of a principal
bundle $P\rightarrow M$, if it is invariant under the principal action of $P$,
naturally reduces to a metric together with a gauge field on the base manifold
$M$. We propose a generalization of this Kaluza-Klein principle to higher
principal bundles and higher gauge fields. For the particular case of the
abelian gerbe of Kalb-Ramond field, this Higher Kaluza-Klein geometry provides
a natural global formulation for Double Field Theory (DFT). In this framework
the doubled space is the total space of a higher principal bundle and the
invariance under its higher principal action is exactly a global formulation of
the familiar strong constraint. The patching problem of DFT is naturally solved
by gluing the doubled space with a higher group of symmetries in a higher
category. Locally we recover the familiar picture of an ordinary para-Hermitian
manifold equipped with Born geometry. Infinitesimally we recover the familiar
picture of a higher Courant algebroid twisted by a gerbe (also known as
Extended Riemannian Geometry). As first application we show that on a
torus-compactified spacetime the Higher Kaluza-Klein reduction gives
automatically rise to abelian T-duality, while on a general principal bundle it
gives rise to non-abelian T-duality. As final application we define a natural
notion of Higher Kaluza-Klein monopole by directly generalizing the ordinary
Gross-Perry one. Then we show that under Higher Kaluza-Klein reduction, this
monopole is exactly the NS5-brane on a $10d$ spacetime. If, instead, we smear
it along a compactified direction we recover the usual DFT monopole on a $9d$
spacetime.
