  In order to treat multiplicative phenomena in twisted (co)homology, we
introduce a new point-set level framework for parametrized homotopy theory. We
provide a convolution smash product that descends to the corresponding
infinity-categorical product and allows for convenient constructions of
commutative parametrized ring spectra. As an immediate application, we compare
various models for generalized Thom spectra. In a companion paper, this
approach is used to compare homotopical and operator algebraic models for
twisted K-theory.
