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
–categories of presentable
–categories as accessible
localisations of
–categories
of presheaves of parametrised spectra. Applying these results to the homotopy theory of
smooth
–stacks
produces well-behaved (symmetric monoidal) model categories of smooth
parametrised spectra. These models, which subsume previous work of Bunke and
Nikolaus, provide a concrete foundation for studying twisted differential
cohomology.
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