Combinatorial parametrised spectra

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 ∞-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 provide a concrete foundation for studying twisted differential cohomology, and subsume previous work of Bunke and Nikolaus.

Strict algebraic models for rational parametrised spectra II

“…and x * preserve colimits, T is determined by the image of the sphere spectrum T(P) ∼ = T(S) ∧ P. One finds that T(S) ∼ = Σ ∞ + ΩX, with monadic structure corresponding to the concatenation of loops (this can be seen explicitly in certain model category presentations, e.g. [Bra20a,Example 2.20]). The adjunction (x !…”

“…Acknowledgements. This article is the third and final in a series [Bra20a,Bra20b] based on my 2018 PhD thesis. I would like to thank Adam Rohrlach, Hisham Sati and Urs Schreiber for useful comments and encouragement during the writing of this article.…”

“…• With a single exception, we adopt the terminology and notation for parametrised stable homotopy theory used in [Bra20a,Bra20b]. The exception is this: for X-spectra P and Q the Z-graded abelian group of fibrewise stable maps from…”

“…relating X-spectra to differential graded A-modules. We establish this adjunction using the combinatorial models for parametrised spectra studied in [Bra20a] and demonstrate that the adjunction is Quillen for the stable model structure on sequential parametrised spectra. For any X-spectrum P, the X + -coaction on the collapse spectrum X !…”

“…In Section 2 we give a brief ∞-categorical overview of the central constructions of parametrised stable homotopy theory: parametrised spectra, base change functors, fibrewise stabilisations of retractive spaces, and connective covers and Postnikov sections. We often need to work with point-set models for parametrised spectra, for which we recall the sequential parametrised spectra of [Bra20a]. Using these combinatorial models for parametrised spectra we prove a useful technical result characterising fibrewise Eilenberg-Mac Lane spectra parametrised over a connected space X in terms of Z[π 1 X]-modules.…”