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I gave sheaf with transfer an Idea-section
(the entry used to me named “Nisnevich sheaves with transfer”. I have renamed it to singular to stay with our convention and removed the “Nisnevich” from the title, as the concept of transfer as such is really not specific to the Nisnevich topology).
The idea section now is the following. (Experts please complain, and I will try to fine tune further):
Given some category (site) $S$ of test spaces, suppose one fixes some category $Corr_p(S)$ of correspondences in $S$ equipped with certain cohomological data on their correspondence space. Then a sheaf with transfer on $S$ is a contravariant functor on $Corr_p(S)$ such that the restriction along the canonical embedding $S \to Corr_p(S)$ makes the resulting presheaf a sheaf.
Traditionally this is considered for $S$ the Nisnevich site and $Corr_p(S)$ constructed from correspondences equipped with algebraic cycles as discussed at pure motive, (e.g. Voevodsky, 2.1 and def. 3.1.1).
The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of Umkehr map/fiber integration by which the sheaf is not only pulled back along maps, but also pushed forward, hence “transferred” (this concept of course makes sense rather generally in cohomology, see e.g. Piacenza 84, 1.1).
The derived categories those abelian sheaves with transfers for the Nisnevich site with are A1-homotopy invariant provides a model for motives known as Voevodsky motives or similar (Voevodsky, p. 20).
What does it mean “fixes SOME category of correspondences in $S$” ?
These are correspondences equipped with some (co)cycles on their correspondence space. One has to decide which kind of (co)cycles to use.
I do not understand still, what is cocycle ON the correspondence space ? Are you assuming some sort of higher stack theory on correspondence spaces for this ? (or you restrict to abelian case…). Entry correspondence does not give a hint and correspondence space redirects there without being mentioned, so no details on what you might mean is there so far.
For traditional pure motives the correspondence spaces (the “tips” of the correspondence!) are product schemes and a cycle on there (“in there” if you prefer) is an algebraic cycle. More generally one can have other data here. For non-commutative motives the role of the cocycle is played by some bimdule.
I know/understand about the classical case of Voevodsky motives, but I do not understand the axioms of the general formalism as outlined in 1. Besides in that case we have cycles, not cocycles – when do we have cocycles (hence allowing for noncompact objects) ?
First I feel like remarking that I didn’t give an axiom, I just said one can consider (co)coycles on the correspondence space.
The example to keep in mind is that discussed at bivariant cohomology:
let $E$ be an $E_\infty$-ring and write $Pic(E)$ for its Picard $\infty$-groupoid. Then correspondences in the slice $\infty Grpd_{Pic(E)}$ are equivalently correspondences in $\infty Grpd$ which carry a twist $\chi_1$ for $E$-cohomology on their left leg, a twist $\chi_2$ on their right leg, and whose correspondence space is equipped with a (co)cocycle in $(\chi_1,\chi_2)$-twisted bivariant $E$-cohomology. More generally one may consider this after pushing everything down to the point, where $\chi_i$ becomes $E_{\bullet + \chi_1}(X_1)$. $\chi_2$ becomes $E_{\bullet + \chi_2}(X_2)$ and the thing assigned to the tip becomes an $E$-linear map
$E_{\bullet + \chi_1}(X_1) \longrightarrow E_{\bullet+ \chi_2}(X_2) \,.$Now if here the left hand happens to be equivalently $E_\bullet(\ast)$ then this is a cycle in the $\chi_2$-twisted $E$-homology of $X_2$; if the right hand happens to be equivalent to $E_\bullet(\ast)$ then this is equivalently a cocycle in the $\chi_1$-twisted $E$-cohomology of $X_1$.
If here $E = KU$ and the homotopy types involved are those of suitable manifolds, then this (co)cycle can be expressed as a class in KK-theory. Alain Connes had long promoted the idea that classes in KK-theory are to be thought of as K-theoretic analogs of pure motives. Another way to substantiate this (if the above seems not close enough to traditional notions of motives) by Snigdhayan Mahanta, see his pdf slides from the twisted cohomology Workshop in Münster last year.
Thanks for the example, I was thinking to ask about KK-theory. On the other hand, the correspondence space is just a space from the original category. So if you talk about the original category with some sort of structure to have homological cycles then I would expect that I should just make sure that I take a subset of cycles (possibly using the legs to have more structure) such that various things like composition make sense (therefore finiteness assumptions in Voevodsky theory).
Yes, one needs extra structure for composition to make sense. That’s what the note of mine Homotopy-type semantics for quantization shows how to do (joint with Joost Nuiten). I am adding a section now that highlights the analogy with motives further.
Okay, I have now written a first version of a section on motivic structures naturally appearing in linear homotopy-type theory, it’s now in section 7.5 here.
I’d be grateful to any motivic expert who goes and has a look and lets me know of his most immediate complaints on my possibly inaccurate use of termionology at some points, or whatever else seems worthy of complaint. I’ll try to fine tune then. Besides complaints, I will also be open to comments such as: “That’s a really cool perspective!” :-)
take a subset of cycles (possibly using the legs to have more structure) such that various things like composition make sense (therefore finiteness assumptions in Voevodsky theory).
Voevodsky uses finite correspondences because they are the largest class for which push-forward and pull-back are well-behaved (without having to work up to rational equivalence as with pure motives). (By the way, in derived algebraic geometry one can just use all the correspondences).
Urs, I look forward to reading this section. I get a little lost when you talk about Umkehr maps and such…
Edit: ah great, will take a look later tonight.
Thanks, Adeel, I’d be really interested in your input here. The section that comments on the relation to motives is now section 7.5 here.
The pull-push via Umkehr maps is abstractly axiomatized in section 4.4 there. I think the axiomatization is actually pleasingly transparent. To me it’s one of the continuing sources of pleasure that in homotopy-type theory the really deep concepts have simpler and more transparent expressions than one is used to from literature on the structures that model this. So maybe give it a try.
Then section 6.2 is meant to show how the abstract axioms serve to encode actual pull-push in twisted generalized cohomology. I am being a bit brief there in that I am being lazy and just giving enough details so that one can make the translation to Joost Nuiten’s thesis, which has more details. But with a little luck I’ll find the time to expand section 6.2.
Also check out section 6.1 which is a super-simple toy example, but maybe all the more serves to give an insight into what’s going on (I hope it does, if not, let me know and maybe I can do something about it).
First just some typos: “defintion” on page 47 and “funcotrs” on page 48.
Edit: Also “Corrrespondences” in the title of section 4.4 (page 26), and “duch” on page 37.
As for the actual content, the analogy is very interesting. Just a few superficial comments…
This form of a datum given by a correspondence of spaces equipped with cycles on the correspondence space is familiar from the definition of pure motives (in particular Chow motives), where the spaces considered are smooth schemes and the cocycles are algebraic cycles (e.g. section 2.1 in [Vo00]).
You write “pure motives” and “Chow motives” but then refer to Voevodsky’s paper where he constructs mixed motives. For your purposes I think it doesn’t matter whether you consider pure motives or mixed motives, so probably you could just change “pure motives” to “mixed motives”, or alternatively “smooth schemes” to “smooth projective schemes” and refer to something about Chow motives, like one of the following
Yuri Manin, Correspondences, motifs and monoidal transformations, Math. USSR Sb. 6 439, 1968
Michel Demazure, Motifs des varieties algebriques, Seminaire N. Bourbaki, 1969-1970, exp. no 365, p. 19-38.
Tony Scholl, Classical motives, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187
Regarding the remark on noncommutative motives, personally I view this theory as not more general, but rather capturing different kind of information than commutative motives.
Regarding the reference to A^1-homotopy theory at the beginning, I think this is not the most accurate term to use here. Though the construction of the A^1-homotopy category, or motivic stable homotopy category, is similar to the constructions of the categories of pure and mixed motives, one main difference is that correspondences don’t really come into A^1-homotopy theory. Similarly at the end,
…are known as sheaves with transfer in motivic homotopy theory. In the seminal model of motivic cohomology given by Voevodsky, the infinity-category of sheaves with transfer that only see the underlying homotopy-type of the geometric spaces (the A^1-homotopy type for smooth schemes) constitutes a realization of the motivic homotopy category (e.g. section 3.1 of [?]).
I have the same issue with the references to motivic homotopy theory. Also the reference to motivic cohomology seems to come out of nowhere, and I wonder what the reference [?] is supposed to be?
Thanks for these reactions!
Some questions and re-reactions, to make sure:
Regarding noncommutative motives; I was thinking of the relation between Chow motives and NCG Chow motives as recalled in Tabuada 11, theorem 4.6 which says that Chow motives fully faithfully embed into noncommutative Chow motives after factoring out the Tate motive.
Regarding $\mathbb{A}^1$-homotopy and correspondences: I was thinking of the construction via $\mathbb{A}^1$-homotopy invariant sheaves with transfer, def. 3.1.10 and then p. 20 in Voevodsky’s “Triangulated categories of motives over a field” (pdf).
For my purpose I don’t need to highlight the relation to $\mathbb{A}^1$-homotopy. What I need is that the homotopy category of motives consists of sheaves with transfer. But it seems to me for that to be true it has to be $\mathbb{A}^1$-homotopy invariant sheaves.
I do realize that the first point here requires saying “Chow motive” while the second uses “mixed motive”. Is that a real technical problem or more a matter of which focus these specific reference take?
Finally regarding the three references that you give at the beginning: I had looked at the first one, not at the other two yet. I am still after a reference that would seem to really be a good introduction. I just want to cite one, not three, since all this is more of a side remark already, for my purposes. If you had to pick one single reference which you would hand to outsiders of motivic theory to read as an introduction on what the field is about, what would be your preferred best choice? Which reference do you wish somebody had pointed you to first as a student when you were beginning to study motives?
I have now polished that section 7.5 a bit more.
I saw only later that a citation came out as a question mark, due to a typo of mine, and that you were wondering about that. This citation was crucially that article by Voevodsky “Triangulated categories of motives over a field” (pdf) where he constructs a version of $DM$ as the category of ($\mathbb{A}^1$-homotopy invariant) sheaves with transfer.
Maybe there is a better reference for this kind of construction? What I am after here is the striking conceptual similarity between motives and QFTs-via-quantization in that both (the former via that construction as sheaves with transfer) are abelian-stuff-valued functors on categories of correspondences-with-cycle-data.
Back to your comments above, one of my paragraphs I have now reworked to read as follows
This form of a datum given by a correspondence of spaces equipped with cycles on the correspondence space is familiar from the definition of (pure or mixed) motives, where the spaces considered are (projective) smooth schemes and the cocycles are algebraic cycles (see section 8 of chapter 1 of [Connes et al’s book] for a review). Moreover, there is a generalization of (pure Chow) motives to non-commutative motives (see [Tabuada’s survey] around theorem 4.6 for a survey) where the role of the cycles is played by bimodules over algebraic data associated with the left and the right leg of the correspondence.
Does that sound reasonable to your ear now?
I think it’s just a terminology confusion: normally, A^1-homotopy theory (also called motivic homotopy theory) refers to the stable motivic homotopy category SH, which is the stabilization of the category of motivic spaces, in analogy with the classical stable homotopy category, the stabilization of the category of topological spaces. This was introduced in the paper
It seems that you are using the term “A^1-homotopy category” rather for the triangulated category of mixed motives DM, as constructed in the paper of Voevodsky you refer to. I guess the confusion arises from the use of A^1-homotopy invariant sheaves in the construction of DM.
Regarding noncommutative motives; I was thinking of the relation between Chow motives and NCG Chow motives as recalled in Tabuada 11, theorem 4.6 which says that Chow motives fully faithfully embed into noncommutative Chow motives after factoring out the Tate motive.
This is true, with rational coefficients, but the better way to put it is that noncommutative motives are a generalization of “K-motives” (karoubian envelope of the category of K_0-correspondences), and K-motives are the same as Chow motives modulo Tate twists just because Grothendieck-Riemann-Roch gives an isomorphism between the Chow group and the Grothendieck group in rational coefficients. Conceptually, Chow motives capture information about Weil cohomologies, while K-motives or noncommutative motives capture information about K-theory, cyclic homology, etc. So I don’t think motivic experts would like the term “generalization” here.
I do realize that the first point here requires saying “Chow motive” while the second uses “mixed motive”.
Sorry, which are the first and second points you mean here?
If you had to pick one single reference which you would hand to outsiders of motivic theory to read as an introduction on what the field is about, what would be your preferred best choice? Which reference do you wish somebody had pointed you to first as a student when you were beginning to study motives?
Of the three references on pure/Chow motives I mentioned, probably Tony Scholl’s is the most readable, though I only discovered it after reading Demazure’s notes. However now I’ve just remembered that Levine has very nice notes
which cover both pure and mixed motives. This may be the best reference, actually.
Thanks!
Yes, I’ll point to Marc Levine’s lectures, that’s a great reference, thanks for reminding me of that.
And thanks for mentioning K-motives. That’s of course all the better for the point I am trying to make. Could you give me a good citation for these? What’s the precise definition? (Is this in Tabuada’s review? I need to check again.)
And on $\mathbb{A}^1$-homotopy: hm, so maybe I don’t want to say “unstable” here, since $DM^{eff}_{gm}$ is stable (triangulated). I could just leave away the reference to $\mathbb{A}^1$-homotopy. But is that really so that the localization of $K^b(Cor_{fin})$ at $\mathbb{A}^1$-homotopy equivalences that yields $DM^{eff}_{gm}$ is not referred to in any way using the words “$\mathbb{A}^1$-homotopy theory”? Well, I could just leave away the word “theory”, maby that makes it better.
Anyway, thanks again. I’ll try to incorporate the aspect of K-motives, that is good.
I don’t really know a good reference for them, but they are covered in Tabuada’s paper “Chow motives versus noncommutative motives”.
But is that really so that the localization of $K^b(Cor_{fin})$ at $\mathbb{A}^1$-homotopy equivalences that yields $DM^{eff}_{gm}$ is not referred to in any way using the words “$\mathbb{A}^1$-homotopy theory”? Well, I could just leave away the word “theory”, maby that makes it better.
That’s right, even though this process is usually called “A^1-localization”. (Personally I don’t like the term “A^1-homotopy theory” at all.) I think I would just say “the theory of mixed motives” wherever you say “A^1-homotopy theory”.
I have edited that section 7.5 a bit further. Probably better I’ll polish it further still tomorrow when I am more awake (should that happen…), but thanks to your very helpful pointer to K-motives I could now make the relation between motives and prequantum integral kernels in linear homotopy-type theory more precise by observing that we have the following zig-zag of comparison functors:
$\array{ Chow\;motives \\ \uparrow^{\mathrlap{Chern\; character}} \\ K-motives \\ \downarrow^{\mathrlap{Tabuada}} \\ noncommutative\;motives \\ \uparrow^{\mathrlap{Mahanta}} \\ KK-classes \\ \uparrow^{\mathrlap{Nuiten}} \\ prequantum\;integral\;kernels }$That is a pretty nice diagram! By the way, perhaps it should be pointed out that the equivalence between K-motives and Chow motives modulo Tate twists in rational coefficients was first noted by Kontsevich in “Notes on motives in finite characteristic”, near the end… I’m on my phone but can add the precise reference later.
cross-linked sheaf with transfer with Mackey functor, pointing to
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