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nLab > Latest Changes: model structure on chain complexes

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTim_Porter
    • CommentTimeOct 16th 2014
    • (edited Oct 16th 2014)

    At model structure on chain complexes, an ’anonymous editor’ suggests that a line saying ’blah blah’ should be completed to something more illuminating!

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 17th 2014
    • (edited Oct 17th 2014)

    Just for reader’s convenience: this refers to this precise spot in the entry.

    • CommentRowNumber3.
    • CommentAuthorTim_Porter
    • CommentTimeOct 17th 2014

    (I just fixed a typo.)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2020

    added pointer to

    diff, v64, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 21st 2020
    • (edited Aug 21st 2020)

    added references which claim (but don’t prove) the projective model structure on connective cochain complexes:

    The projective model structure on connective cochain complexes is claimed, without proof, in:

    diff, v64, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2021

    briefly added the statement (here) that the projective model structure on connective chain complexes is monoidal.

    diff, v73, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2021
    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2021
    • (edited Jul 21st 2021)

    Added (here) statement and proof, following Prop. 24 in

    (which I added), that the model structures with (co)fibrations the degreewise injections/surjections are proper.

    No proof yet that this works for bounded chain complexes with conditions in positive degrees.

    diff, v77, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2023

    have polished-up the list of the first few references and their commentary in References – For unbounded chain complexes (here).

    In particular I have moved up the item

    to after the other references for proofs of cofibrant generation, highlighting that, in addition, this contains proof of properness and monoidalness.

    Finally, I added publication data to this item:

    diff, v80, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 7th 2023

    I am wondering about the following:

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

    And then:

    For 𝒮∈sGrpd\mathcal{S} \in sGrpd an object in the model structure on simplicial groupoids

    and for sFunc(𝒮,sCh •)sFunc(\mathcal{S}, s\mathrm{Ch}_\bullet) the (projective, probably) model structure on simplicial functors,

    how good is the model structure on the Grothendieck construction ∫ 𝒮sFunc(𝒮,sCh •)\int_{\mathcal{S}} sFunc(\mathcal{S}, s\mathrm{Ch}_\bullet) ?

    For instance: Might this still be right proper?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023
    • (edited Apr 18th 2023)

    noticed that these two versions are substantially different, so I split this up into two items:

    diff, v83, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023
    • (edited Apr 18th 2023)

    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)

    diff, v83, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

    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.

    diff, v85, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2023

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

    diff, v86, current

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 18th 2023

    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.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023
    • (edited Apr 19th 2023)

    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.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023
    • (edited Apr 19th 2023)

    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

    sGrpd ⟶ ModCat 𝒳 ↦ sFunc(𝒳,sCh •(k) proj) inj \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:𝒳→𝒳′f : \mathcal{X} \to \mathcal{X}' we have that f !f_! preserves all weak equivalences.

    I was thinking of arguing this by observing that on systems of chain complexes of vector spaces, f !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

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

    2. and then objectwise to the simplicial functor category sFunc(𝒳,sCh •(k)) injsFunc\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 !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 !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.

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 19th 2023

    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.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2023

    Thanks!

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

    • CommentRowNumber20.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 19th 2023

    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.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023
    • (edited Apr 20th 2023)

    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 HℂH \mathbb{C}-module spectra over varying bases, i.e. of the ∞\infty-Grothendieck construction

    ∫𝒳∈Grpd ∞HℂMod 𝒳, \underset{\mathcal{X} \in Grpd_\infty} \int H \mathbb{C} Mod_{\mathcal{X}} \,,

    i.e. of the “HℂH \mathbb{C}-linear” version of the tangent ∞\infty-topos of Grpd ∞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 HℂH\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 HℂH\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(−,sCh •(ℂ))sFunc( - , s Ch_\bullet(\mathbb{C})) above.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2023

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

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023
    • (edited Apr 21st 2023)

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

    I know that the plain Reedy structure on sCh •(k)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]

    • CommentRowNumber24.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 21st 2023
    • (edited Apr 21st 2023)

    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.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2023

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

    • CommentRowNumber26.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 21st 2023

    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.

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2023
    • (edited Apr 23rd 2023)

    Back to the cofibrancy issue:

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

    diff, v93, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2023
    • (edited Apr 23rd 2023)

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

    diff, v93, current

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2023

    I have forwarded the question about monoidal structure on sCh •sCh_\bullet to MathOverflow: MO:q/445397

    • CommentRowNumber30.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 24th 2023

    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.

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023

    Taking a step back, I have first of all added now the statement that unbounded Ch •(RMod)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 RR any integral ideal domain.

    diff, v95, current

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023
    • (edited Apr 26th 2023)

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

    diff, v95, current

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeApr 26th 2023

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

    diff, v98, current

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2023

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

    diff, v101, current

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMay 8th 2023

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

    diff, v104, current

1 to 35 of 35

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