  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.
