Twisted Parametrized Stable Homotopy Theory

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams of simplicial categories. We present a classification of bundles of categories of spectra and of the associated twisted parametrized spectra. Though twisted parametrized spectra do not have global homotopy types and therefore do not have generalized homology invariants in the usual sense, they do admit generalized-homology-type invariants for certain commutative ring spectra. We describe this invariant theory and in particular note that under mild hypotheses, a twisted parametrized spectrum will have cyclically graded homology invariants and will also have K-theory and complex bordism invariants, as expected for a Floer or semi-infinite homotopy type. We discuss the association of a twist of parametrized homotopy theory to a polarized infinite-dimensional manifold and present a conjectural, explicit realization of this twist in terms of parametrized semi-infinitely indexed spectra.