nLab model structure for parameterized spectra

Idea

On model category structures presenting $\infty$-categories of parameterized spectra (in parameterized stable homotopy theory), either over a fixed base or over varying bases and then presenting tangent $\infty$-categories.

(…)

Properties

We consider some properties of model category

(1)$$\mathrm{Sp}^\Sigma_{\mathcal{R}} \;\; \in \;\; ModCat$$

of parameterized symmetric spectra in $\mathcal{I}$-spaces based on SimplicialSets and equipped with the absolute/positive local model structures.

$$\mathcal{S} \coloneqq sSet$$

as given in Hebestreit, Sagave & Schlichtkrull (2020):

Proposition

With the external smash product of spectra the model category (1)

1. is a symmetric closed monoidal category

2. with tensor unit the sphere spectrum $\mathcal{S}$,

3. which satisfies the pushout-product axiom

4. and the unit axiom,

hence is a monoidal model category.

Proof

The first statements are made explicit in HSS, Prop. 4.5 & Prop. 5.11. To see that also the unit axiom is satisfied it is sufficient to see that $\mathbb{S}$ is a cofibrant object. For that use the definition of $\mathbb{S}$ as $\mathbb{S}_t^{\mathcal{I}}[\ast]$ (hidden on p. 38) and the fact that $\mathbb{S}_t^{\mathcal{I}}$ is a left Quillen functor, by HSS, Lem. 5.12.

Proposition

The inclusion of $\mathcal{I}$-spaces via their zero-spectrum bundles is a bireflective subcategory (according to this Def.) which is also a Quillen adjoint triple:

(2)

Proof

This may be recognized from HSS, Lem. 3.19 & Cor. 5.13.

Definition

In the following we will leave the inclusions

$$\mathcal{S} \overset{const}{\hookrightarrow} \mathcal{S}^{\mathcal{I}} \overset{0}{\hookrightarrow} \mathrm{Sp}^\Sigma_\mathcal{R}$$

notationally implicit. For example, for $B$ a simplicial set we write $\big(\mathrm{Sp}^\Sigma_\mathcal{R}\big)_{/B}$ for the slice model structure of (1) over the zero-spectrum bundle over $B$.

The model category (1) is not quite right proper (cf. pp. 40) but, in its version based on simplicial sets, we have:

Proposition

Left base change $f^\ast$ along Kan fibrations $f \,\colon\, B_1 \to B_2$ of (zero-spectrum bundles (2) over) Kan complexes is a left Quillen functor between the slice model structures of the absolute local model structure. Since, generally, $f^\ast$ is also a right Quillen functor, we have base change Quillen adjoint triples of this form:

Proof

The left Quillen property of $f^\ast$ is HSS20, Lem. 7.22](#HebestreitSagaveSchlichtkrull20). The right Quillen property is mentioned inside the proof of Lem. 8.10 there.

Similarly, but now also for the positive local model structure, we have:

Proposition

With respect absolute or positive local model structure:

Proof

HSS, Lem. 5.17 & Lem. 7.25

(…)

Related concepts

• model structure for spectra

• model structure on presheaves of spectra

• parameterized stable homotopy theory

• tangent (infinity,1)-category

References

Tools

Tools for creating model structures on spectra relative to suitable ambient model categories:

• Stefan Schwede, Section 3 of: Spectra in model categories and applications to the algebraic cotangent complex, Journal of Pure and Applied Algebra 120 (1997) 104 [pdf]

and also for model structures on symmetric spectra:

• Mark Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 1 (2001) 63-127 [arXiv:0004051, doi:10.1016/S0022-4049(00)00172-9]

Over a fixed base

Model structure for genuine equivariant spectra parameterized over a fixed base:

• Peter May, Johann Sigurdsson, Chapter 12 of: Parametrized Homotopy Theory, Mathematical Surveys and Monographs 132, AMS (2006) [arXiv:math/0411656]

Over varying bases

Model structures of parameterized spectra over varying bases:

• Vincent Braunack-Mayer, Combinatorial parametrised spectra, Algebr. Geom. Topol. 21 (2021) 801-891 [arXiv:1907.08496, doi:10.2140/agt.2021.21.801]

  (based on the PhD thesis, 2018)

• Fabian Hebestreit, Steffen Sagave, Christian Schlichtkrull, Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces, Forum of Mathematics, Sigma 8 (2020) e16 [arXiv:1904.01824, doi:10.1017/fms.2020.11]