nLab model structure for parameterized spectra



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

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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.



We consider some properties of model category

(1)Sp ΣModCat \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.

𝒮sSet \mathcal{S} \coloneqq sSet

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


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 𝒮\mathcal{S},

  3. which satisfies the pushout-product axiom

  4. and the unit axiom,

hence is a monoidal model category.


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 𝕊\mathbb{S} is a cofibrant object. For that use the definition of 𝕊\mathbb{S} as 𝕊 t [*]\mathbb{S}_t^{\mathcal{I}}[\ast] (hidden on p. 38) and the fact that 𝕊 t \mathbb{S}_t^{\mathcal{I}} is a left Quillen functor, by HSS, Lem. 5.12.


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:



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


In the following we will leave the inclusions

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

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

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


Left base change f *f^\ast along Kan fibrations f:B 1B 2f \,\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 *f^\ast is also a right Quillen functor, we have base change Quillen adjoint triples of this form:


The left Quillen property of f *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:


With respect absolute or positive local model structure:




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:

Over a fixed base

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

Over varying bases

Model structures of parameterized spectra over varying bases:

Last revised on April 20, 2023 at 08:54:10. See the history of this page for a list of all contributions to it.