  We obtain combinatorial model categories of parametrised spectra, together
with systems of base change Quillen adjunctions associated to maps of parameter
spaces. We work with simplicial objects and use Hovey's sequential and
symmetric stabilisation machines. By means of a Grothendieck construction for
model categories, we produce combinatorial model categories controlling the
totality of parametrised stable homotopy theory. The global model category of
parametrised symmetric spectra is equipped with a symmetric monoidal model
structure (the external smash product) inducing pairings in twisted cohomology
groups.
  As an application of our results we prove a tangent prolongation of Simpson's
theorem, characterising tangent $\infty$-categories of presentable
$\infty$-categories as accessible localisations of $\infty$-categories of
presheaves of parametrised spectra. Applying these results to the homotopy
theory of smooth $\infty$-stacks produces well-behaved (symmetric monoidal)
model categories of smooth parametrised spectra. These models provide a
concrete foundation for studying twisted differential cohomology, incorporating
previous work of Bunke and Nikolaus.
