  We construct a stable infinity category with objects flow categories and
morphisms flow bimodules; our construction has many flavors, related to a
choice of bordism theory, and we discuss in particular framed bordism and the
bordism theory of complex oriented derived orbifolds. In this setup, the
construction of homotopy types associated to Floer-theoretic data is immediate:
the moduli spaces of solutions to Floer's equation assemble into a flow
category with respect to the appropriate bordism theory, and the associated
Floer homotopy types arise as suitable mapping spectra in this category. The
definition of these mapping spectra is sufficiently explicit to allow a direct
interpretation of the Floer homotopy groups as Floer bordism groups. In the
setting of framed bordism, we show that the category we construct is a model
for the category of spectra. We implement the construction of Floer homotopy
types in this new formalism for the case of Hamiltonian Floer theory.
