  The integration problem of a C-bracket and a Vaisman (metric, pre-DFT)
algebroid which are geometric structures of double field theory (DFT) is
analyzed. We introduce a notion of a pre-rackoid as a global group-like object
for an infinitesimal algebroid structure. We propose that several realizations
of pre-rackoid structures. One realization is that elements of a pre-rackoid
are defined by cotangent paths along doubled foliations in a para-Hermitian
manifold. Another realization is proposed as a formal exponential map of the
algebroid of DFT. We show that the pre-rackoid reduces to a rackoid that is the
integration of the Courant algebroid when the strong constraint of DFT is
imposed. Finally, for a physical application, we exhibit an implementation of
the (pre-)rackoid in a three-dimensional topological sigma model.
