model category, model $\infty$-category
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Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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vector bundle, 2-vector bundle, (∞,1)-vector bundle
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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
of parameterized symmetric spectra in $\mathcal{I}$-spaces based on SimplicialSets and equipped with the absolute/positive local model structures.
as given in Hebestreit, Sagave & Schlichtkrull (2020):
With the external smash product of spectra the model category (1)
with tensor unit the sphere spectrum $\mathcal{S}$,
which satisfies the pushout-product axiom
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 $\mathbb{S}$ 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.
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
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:
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:
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:
With respect absolute or positive local model structure:
(…)
Tools for creating model structures on spectra relative to suitable ambient model categories:
and also for model structures on symmetric spectra:
Model structure for genuine equivariant spectra parameterized over a fixed base:
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]
Last revised on April 20, 2023 at 08:54:10. See the history of this page for a list of all contributions to it.