A parametrized spectrum
is a
family of spectra
continuously
parametrized by the points
of a topological space. We take the point of view that a
parametrized spectrum is a bundle-theoretic geometric object. When
is a ring spectrum, we consider
parametrized
-module
spectra and show that they give cocycles for the cohomology theory determined by the algebraic
-theory
of
in a manner analogous to the description of topological
-theory
as the Grothendieck group of vector bundles over
. We prove
a classification theorem for parametrized spectra, showing that parametrized spectra over
whose fibers are equivalent
to a fixed
-module
are classified by homotopy classes of maps from
to the classifying space
of the topological
monoid of
-module
equivalences from
to
.
