# Combinatorial parametrised spectra

## TL;DR

This paper develops combinatorial model categories for parametrised spectra, establishing a framework for twisted cohomology and smooth $ty$-stacks, with applications to tangent $ty$-categories and differential cohomology.

## Contribution

It introduces a comprehensive combinatorial model for parametrised spectra with base change adjunctions and symmetric monoidal structures, advancing the foundations of parametrised stable homotopy theory.

## Key findings

- Constructed combinatorial model categories for parametrised spectra.
- Established symmetric monoidal model structures for global parametrised spectra.
- Applied the framework to tangent ty-categories and twisted differential cohomology.

## Abstract

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.