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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).
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.
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