Combinatorial parametrised spectra

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