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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 31st 2023
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   Author: Urs
   Format: MarkdownItexstarting something, but just a couple of references so far <a href="https://ncatlab.org/nlab/revision/model+structure+for+parameterized+spectra/1">v1</a>, <a href="https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra">current</a>

   starting something, but just a couple of references so far

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2023
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   Author: Urs
   Format: MarkdownItexmade a note ([here](https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra#QuillenRightBaseChange)) that: > The model category $Sp^\Sigma_{\mathcal{R}}$ of parameterized spectra given in [Hebestreit, Sagave & Schlichtkrull (2020)](#HebestreitSagaveSchlichtkrull20) is not quite [[right proper]] (cf. pp. 40) but, in its version based on [[simplicial sets]], left [[base change]] $f^\ast$ along [[Kan fibrations]] $f \,\colon\, B_1 \to B_2$ of ([[zero-spectrum bundles]] over) [[Kan complexes]] *is* a [[left Quillen functor]] between the [[slice model structures]] (by [HSS20, Lem 7.22](#HebestreitSagaveSchlichtkrull20)). <a href="https://ncatlab.org/nlab/revision/diff/model+structure+for+parameterized+spectra/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+for+parameterized+spectra/4">v4</a>, <a href="https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra">current</a>

   made a note (here) that:

   The model category $Sp^\Sigma_{\mathcal{R}}$ of parameterized spectra given in Hebestreit, Sagave & Schlichtkrull (2020) is not quite right proper (cf. pp. 40) but, in its version based on simplicial sets, left base change $f^\ast$ along Kan fibrations $f \,\colon\, B_1 \to B_2$ of (zero-spectrum bundles over) Kan complexes is a left Quillen functor between the slice model structures (by HSS20, Lem 7.22).

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2023
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   Author: Urs
   Format: MarkdownItexUnder "Properties" ([here](https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra#Properties)) I have started a list with some facts extracted from [Hebestreit, Sagave & Schlichtkrull (2020)](https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra#HebestreitSagaveSchlichtkrull20). I'd like to conclude that the last base change Quillen adjoint triple generalizes to module spectra. This should follow immediately if the monoid axiom holds in the positive local model structure based on simplicial sets. That this is the case seems to be at least *implicit* in their text, such as from the last line on [p. 30](https://arxiv.org/pdf/1904.01824.pdf#page=30) (which laments that the monoid axioms fails with respect to topological spaces) -- but I am not sure. <a href="https://ncatlab.org/nlab/revision/diff/model+structure+for+parameterized+spectra/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/model+structure+for+parameterized+spectra/6">v6</a>, <a href="https://ncatlab.org/nlab/show/model+structure+for+parameterized+spectra">current</a>

   Under “Properties” (here) I have started a list with some facts extracted from Hebestreit, Sagave & Schlichtkrull (2020).

   I’d like to conclude that the last base change Quillen adjoint triple generalizes to module spectra. This should follow immediately if the monoid axiom holds in the positive local model structure based on simplicial sets. That this is the case seems to be at least implicit in their text, such as from the last line on p. 30 (which laments that the monoid axioms fails with respect to topological spaces) – but I am not sure.

   diff, v6, current