spelled out in more detail (here) how to reduce to that monoidal localization criterion by Barwick

]]>made explicit (here) the set of generating cofibrations for the unbounded projective strcuture

]]>to the proposition and proof about existence and Quillen equivalence of $sCh_\bullet$ I added the explicit remark that it is also still combinatorial (here)

]]>added (here) the statement of monoidalness of $Ch_\bullet(k) \simeq_{Qu} sCh_\bullet(k)$, for the moment just with a pointer to the MO-discussion

]]>Taking a step back, I have first of all added now the statement that unbounded $Ch_\bullet(R Mod)$ is monoidal model itself, in the first place: here.

This ought to be classical, though I am not sure what to reference. An explicit claim appears in Strickland (2020), Prop. 25 which applies, if I understand well, for $R$ any integral ideal domain.

]]>Re #25: Concerning the remark about cellularity, Hirschhorn’s proof only makes use of it in the other parts of the theorem. The proof of Part 4 (including its dependencies) does not use cellularity.

]]>I have forwarded the question about monoidal structure on $sCh_\bullet$ to MathOverflow: MO:q/445397

]]>Finally added the statement (here) that also all objects of the simplicial enhancement $sCh_\bullet(k Mod)$ of the projective unbounded model structure are cofibrant over a ground field.

]]>Back to the cofibrancy issue:

Have added the argument here that every unbounded chain complex of vector spaces is projectively cofibrant.

]]>Re #25: I am not sure why you are saying that Barwick’s paper, published in HHA, is unpublished. A precise reference is given in the article.

]]>But Barwick’s seems to be an unpublished preprint (?) and Hirschhorn discusses the cellular situation, not the combinatorial situation.

]]>Re #23: I am not sure why you say that Lurie’s version is the “usual theorem”.

The standard source for left Bousfield localizations is Barwick (On left and right …) and Hirschhorn.

Their theorems are stated without simplicial enrichments. However, if the original model category is simplicial, the so is its left Bousfield localization, see Hirschhorn, Theorem 4.1.1, Part 4.

]]>One point I am still stuck on is seeing that the model structure on $sCh_\bullet(k)$ (here) is combinatorial, to ensure that the injective model structure on $sFunc\big(\mathcal{X}, sCh_\bullet(k)\big)$ exists.

I know that the plain Reedy structure on $sCh_\bullet(k)$ is left proper and combinatorial, but I need to see that this remains true for its left Bousfield localization. Here the issue is that the plain Reedy structure is definitely not simplicial, so that the usual theorem for combinatorial Bousfield localizations does not apply.

On the other hand, it is close to simplicial, by RSS’s Cor. 7.4 (p. 15), maybe that’s enough?

[edit: Or maybe we don’t need simplicial enrichment? that’s a gap in my education: Barwick here states Smith’s theorem without that assumption]

]]>For what it’s worth, I have completed (I think) and typed out in more detail the above proof that $\underset{\mathcal{X} \in sGrpd}{\int} sFunc\big( \mathcal{X} ,\, sCh_\bullet(k) \big)$ carries the “integral” model structure: currently in the Sandbox.

]]>True, maybe there are better ways to construct this, I’ll be happy to try whatever works.

What I am after is a good model category presentation of the $\infty$-category of parameterized $H \mathbb{C}$-module spectra over varying bases, i.e. of the $\infty$-Grothendieck construction

$\underset{\mathcal{X} \in Grpd_\infty} \int H \mathbb{C} Mod_{\mathcal{X}} \,,$i.e. of the “$H \mathbb{C}$-linear” version of the tangent $\infty$-topos of $Grpd_\infty$ (which is still an $\infty$-topos, by Joyal locus-yoga).

Here by “good” I ultimately mean some kind of type-theoretic model category compatibly equipped with a classical modality and with external-smash monoidal model structure, all so that it can interpret the linear homotopy type theory of Riley 2022. But I’ll be happy with less good model structures to start with.

Another approach I was looking into is to start with existing external-smash monoidal model structures for parameterized spectra over varying bases (here) and then try to pass to the associated model structure of $H\mathbb{C}$-modules internal to these. The monoid axiom should hold here, though the literature is a little unclear (as I commented in another thread here).

But even if this works it will probably be clunky to use in practice. For applicability I’d much rather make use of the stable Dold-Kan correspondence and realize the parameterized $H\mathbb{C}$-module spectra as $\infty$-groupoid-parameterized chain complexes of complex vector spaces (“$\infty$-local systems”) right away. Whence my attempt to get the Grothendieck construction model category of $sFunc( - , s Ch_\bullet(\mathbb{C}))$ above.

]]>Re #17: Before I respond to the question proper, allow me to clarify the underlying context: what ∞-categorical idea are you trying to encode?

For example, given that you consider simplicial functors from simplicial groupoids to chain complexes, you might as well observe that for every pair of objects, the induced map hom(x,y)→hom(Fx,Fy) lands in a module over HZ, so by adjunction, you might as well write down a map NZ[hom(x,y)]→Hom_Ch(Fx,Fy), and work exclusively with chain complexes, avoiding simplicial sets and simplicial chain complexes altogether.

]]>Thanks!

I have added the Barwick reference there at “monoidal localization”. Will need to absorb this…

]]>Of course, your argument might apply in all of these cases. Now I realize that I don’t know the sufficient conditions for “monoidal Bousfield localization”. What’s a source for this?

Yes, the argument also works just fine for the Reedy model structure, since it is left proper and combinatorial. Most properties, including left/right properness, only depend on weak equivalences.

Concerning monoidality of Bousfield localizations: see Barwick’s paper On left and right…, Proposition 4.47, where he formulates the same condition in the adjoint form. The nLab has the article monoidal localization, and the version for Bousfield localizations is a straightforward generalization.

]]>By the way, regarding my second question from #10, I am trying to proceed as follows:

One sticking point is to show that the functor

$\array{ sGrpd &\longrightarrow& ModCat \\ \mathcal{X} &\mapsto& sFunc\big(\mathcal{X}, sCh_\bullet(k)_{proj}\big)_{inj} }$is “left proper” in the sense of Harpaz & Prasma 2015 p. 17, hence that for every acyclic cofibration $f : \mathcal{X} \to \mathcal{X}'$ we have that $f_!$ preserves all weak equivalences.

I was thinking of arguing this by observing that on systems of chain complexes of *vector spaces*, $f_!$ essentially already coincides with its derived functor, as follows:

Namely, by this Prop., every bounded-below chain complex of vector spaces is cofibrant, whence every chain complex of vector spaces is a sequential colimit of cofibrant objects (namely over all its co-connective stages), and over a diagram whose morphisms are cofibrations. I think.

Moreover, this sequential decomposition is clearly functorial, so that every quasi-isomorphism between chain complexes is a corresponding sequential colimit, in the arrow category, of quasi-isomorphisms between bounded-below chain complexes.

Now I want to argue that these arguments pass

to the category of simplicial chain complexes (this I need to think more about),

and then objectwise to the simplicial functor category $sFunc\big(\mathcal{X}, sCh_\bullet(k)\big)_{inj}$,

and equipping the latter with the injective model structure means that all these cofibrancy arguments go through objectwise.

Now since $f_!$ commutes with these colimits (being a left adjoint) and takes weak equivalences between cofibrant objects to weak equivalences (being equal to its derived functor here), it follows that $f_!$ applied to any weak equivalence is a colimit of weak equivalences in the arrow category, and since the structure morphisms are cofibrations it is in fact a homotopy colimit of weak equivalence and hence a weak equivalence. QED.

]]>Thanks! I’ll need to think about this.

Could you say which specific version of simplicial enhancement you are considering?

I was thinking of the construction in Rezk, Schwede & Shipley 2001 or Dugger 2001a, but these localize the Reedy model structure, not the projective/injective.

[edit: Sorry, I see now that Dugger 2001a does consider the projective structure, calling is the “Bousfield-Kan structure”.]

[edit 2: Hm, but he seems to prove the simplicial enhancement only for the Reedy version (?)]

Of course, your argument might apply in all of these cases. Now I realize that I don’t know the sufficient conditions for “monoidal Bousfield localization”. What’s a source for this?

I see a relevant Prop. 12.18 in Lawson 2022 but I am not sure how useful that is. You seem to have a stronger statement in mind.

]]>Re #10:

Given that Ch • is proper and monoidal, is the Quillen equivalent simplicial model structure sCh • still proper and monoidal?

sCh is constructed as the left Bousfield localization of the projective or injective model structure on sCh.

The injective model structure on sCh inherits the property of monoidality and left properness from Ch.

Since the class of morphisms with respect to which we take the left Bousfield localization is closed under derived monoidal products with an arbitrary object in Ch, the left Bousfield localization yields a monoidal model category.

It is also automatically left proper.

To answer the question about right properness, apply a theorem of Stenzel: given that sCh with the injective structure is right proper, the left Bousfield localization of sCh is right proper if and only if the localization functor is semi-left exact.

The localization functor L simply takes the homotopy colimit over Δ^op.

Therefore, the question reduces to the following: given homotopy constant objects A,B∈sCh and an arbitrary object C∈sCh, does L preserve the homotopy pullback of the span A→B←C?

Indeed, the answer is positive, since in Ch homotopy colimits over Δ^op distribute over homotopy base changes.

]]>added (here) statement also of the injective model structure on unbounded chain complexes

]]>I have added a remark (here) spelling out the counter-example from Hovey (1999), Rem. 2.3.7 showing that not every unbounded chain complex is projectively co-fibrant, not even in characteristic zero.

]]>to the main proposition asserting the standard model structure on unbounded complexes (here) I have

added the statement that this is combinatorial

and then to the proof I added the argument for why that is (here)

]]>