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for completeness, I created external tensor product of vector bundles
added pointer to:
Maybe you’ve already referenced it, Urs, but there’s a construction of the external smash product of parametrised orthogonal spectra as a left Quillen bifunctor, in Lemma 2.3.11 of Malkiewich’s https://arxiv.org/abs/2305.15327 (which is a disstillation of a previous, longer article on the arXiv).
Thanks. I had not seen this recent preprint yet, but I did look at and reference (here) the precursor (arXiv:1906.04773) that the new one says it is a “condensation” of.
I had put this aside since, as far as I can see, these articles do not consider a global model-category of parameterized spectra over varying base spaces, just the pseudo-functorial system of parameterized spectra over fixed bases. In this base-wise form the external smash product is claimed to be a Quillen bifunctor in Lem 5.4.5 (p. 67) of the new condensation preprint, which was Lem. 6.4.3 in the previous version (p. 96).
Hmm, thanks. It looks like it treats the case of parametrised spectra over arbitrary base, and with morphisms arbitrary commuting (and compatible) squares, so in some sense a Grothendieck construction of the pseudofunctorial system. How does this differ from you expect from a global version? I know it should be something like global equivariant homotopy theory, but I don’t quite know how these two pictures join up, other than “spectrum parametrised by BG” = G-equivariant spectrum.
For the global version we ask for a model $\mathrm{Loc}_{\mathbb{S}}$ of the Grothendieck construction $\int_{\mathcal{X} \in Grpd_\infty} Spectra^{\mathcal{X}}$ (the tangent $\infty$-topos) and then for a Quillen bifunctor on that
$Loc_{\mathbb{S}} \times Loc_{\mathbb{S}} \xrightarrow{\;\; \boxtimes \;\;} Loc_{\mathbb{S}} \,.$Such a global Quillen bifunctor will reduce to systems of Quillen bifunctors over fixed base spaces $\mathcal{X}$, $\mathcal{Y}$
$Spectra^{\mathcal{X}} \times Spectra^{\mathcal{Y}} \xrightarrow{\;\; \boxtimes \;\;} Spectra^{\mathcal{X} \times \mathcal{Y}}$but is a stronger structure. One model for this stronger structure is in Hebestreit, Sagave & Schlichtkrull (2020), which seems quite neat.
I want this for $H\mathbb{K}$-module spectra. To get there from HSS20 I’d need to verify the monoid axiom (it remains unclear to me whether they claim this), then give a model for $H\mathbb{K}$ in their category and then pass to the induced model category of $H\mathbb{K}$-modules, then check that the $H\mathbb{K}$-external tensor is still a Quillen bifunctor there. This might work, but it will be a quite heavy model for what on underlying model categories should be equivalent to $\infty$-local systems over $\mathbb{K}$.
I have been trying to build that global model structure on $\mathbb{K}$-linear $\infty$-local systems directly. So far I get a global external tensor product which is a bit stronger than the base-wise version, but still poor compared to the full version (here).
Oh, I see. So the version in the new preprint misses out what’s going on in the non-vertical morphisms in the Grothendieck construction. Makes sense, thanks!
Hmm, but the paper talks about the left derived functor of the external smash bifunctor defined on the Grothendieck construction, and proves nice things about it at the level of homotopy categories. Getting out of my league here, so I’ll let others take over, or let it lie there.
proves nice things about it
Could you say concretely which paragraph you are referring to here?
For example Theorem 6.1.1, Remark 6.1.4 but I can’t tell if this is what you are hoping for.
I see, thanks. I have added a remark “3.43” (here) comparing to that Remark. 6.1.4.
What would be most useful now: if the monoid axiom were satisfied in that integral model structure $\mathcal{OS}$…
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