Foundation of Floer homotopy theory I: Flow categories

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.