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created motive just in order to link to the sub-pages on this that we already have, and in order to record a link to a useful MO discussion about them.
‘MO’ isn't really a technical term; it referes to Math Overflow, the awesome new resource for professional mathematicians to ask and answer questions. (I sound like an ad, but it is awesome.)
Created Nori's Tannakian theorem. New Kaledin's reference at motive.
Today I quizzed an expert on motives. Finally I got something that I could recognize as a clean formal abstract definition. I have tried to make a quick note that brings out the clean basic idea of the formal definition in a new section motive -- Idea -- Idea of the precise abstract definition.
Experts may still complain, but to my mind this brings the entry lightyears closer to being an actual explanation of what a motive is than it was before.
If you are an expert and still feel appalled by the sheer insufficiency of the entry, maybe the new paragraph at least serves as a condensation point for something better.
Nice!
Pushing monoidal units forwards to a point reminds me of what Kate and I did with indexed monoidal categories. Except that in algebraic-geometry land, they push forward by the right adjoint of pullback…
I wouldn’t denote the initial functor by DM because there already exists an $(\infty,1)$-functor DM which is meant to be initial for a different set of axioms. The usual notation for the initial $(\infty,1)$-functor in the sense meant here is SH. Ideally DM would also satisfy the axioms displayed here (and more), but it’s not known to satisfy the localization property (= property 2).
I think part 2 of the “idea of definition” is wrong: the motive of $X$ would be $p_\sharp 1_X$ where $p_\sharp$ is left adjoint to $p^\ast$.
Hi Marc,
okay, I have changed that. What's your favorite reference that states theses definitions clearly?
I added the word “derived” to that section. This is the idea of derived mixed motives, not the conjectural true abelian category of mixed motives. For pure motives and Nori’s motives one works with nonderived version, but for mixed motives one has only a conjecture, no definition yet.
@Urs: I know of no other reference than Cisinski-Déglise. They define the standard $DM$ in part 3, but they are unable to prove that it satisfies the complete 6-functor formalism. In part 4 they construct a rational version $DM_B$ (should be a cyrillic B) which has the 6-functor formalism. I just quickly went through the updated introduction, and I don’t see them mentioning any universal properties of these 2-functors. However, these 2-functors have obvious “objectwise” universal properties by definition, and the objectwise universal properties together with the 6-functor formalism should be stronger than just a universal property for the 2-functor.
For example, for a fixed base scheme $S$, the functor $Sm/S \to DM(S)$ is the universal functor to a cocomplete stable symmetric monoidal $(\infty,1)$-category which (1) has transfers, (2) satisfies Nisnevich excision, (3) is homotopy invariant, and (4) inverts the Tate object. The rational version $DM_B(S)$ satisfies the same universal property with $\mathbb{Q}$-coefficients and with Nisnevich excision replaced by étale hyperdescent (transfers are then redundant), and various other universal properties if we add regularity assumptions on $S$.
I know of no other reference than Cisinski-Déglise.
Thanks, I see. What about those lecture notes allegedly taken by Deligne of a lecture by Voevodsky, where the idea of these axioms supposedly originates? The article points to Voevodsky’s general IAS web page, which, however, seems to have no trace of a pointer to such notes left (?)
You probably mean these notes. They are superseded by Ayoub’s thesis, but neither of them say anything about DM. Well, they both make it seem as though DM is known to satisfy the axioms (at the very end in Deligne’s notes), but it’s not. As far as I know Cisinski-Déglise has the complete story about DM (and most of it is in the introduction).
You probably mean these notes.
Yes, thanks! I have added that to motive.
They are superseded by Ayoub’s thesis, but neither of them say anything about DM. Well, they both make it seem as though DM is known to satisfy the axioms (at the very end in Deligne’s notes), but it’s not. As far as I know Cisinski-Déglise has the complete story about DM (and most of it is in the introduction).
Thanks. That’s all very useful to know.
I added a whole bunch of modern constructions of the derived category of mixed motives.
I’ve still got a beef with the Idea of the precise abstract definition of derived motives, however. It seems that this section was written under the idea that stable motivic homotopy theory deserves to be called the “derived category of motives”. This is certainly not the case, as all the definitions of the latter that I’ve added should make clear. The analogy is
stable motivic homotopy theory $\leftrightarrow$ stable homotopy theory
derived motives $\leftrightarrow$ chain complexes
(however, the vertical relationship on the LHS is more complicated than on the RHS, even with rational coefficients – see the definition of “Morel motive”).
I suggest moving the content of this section to stable motivic homotopy theory under “six operations”.
I suggest moving the content of this section to stable motivic homotopy theory under “six operations”.
I went ahead and did that: six operations.
I have added to motive in the (References-)section on relation to physics the following:
That the pull-push quantization in Gromov-Witten theory is naturally understood as a “motivic quantization” in terms of Chow motives of Deligne-Mumford stacks was suggested in
Further investigation of these stacky Chow motives then appears in
Bertrand Toën, On motives for Deligne-Mumford stacks, International Mathematics Research Notices 2000, 17 (2000) 909-928 (arXiv:math/0006160, web, pdf) {#Toen00}
Utsav Choudhury, Motives of Deligne-Mumford Stacks (arXiv:1109.5288)
I added a brief paragraph about Deligne’s Tannakian category of mixed motives, with a reference.
Started writing the construction of DM, following Cisinski-Déglise, at
I’ve got up to DM^{eff} for the moment, I think I will add some prerequisites on model structures on spectra, following Ayoub’s thesis, before I do DM.
Also I created
following Déglise.
Thanks!
At some point we may want to merge various things. We have related discussion at motivic homotopy theory and at motive, and maybe elsewhere, too. I am not sure.
I think what is at motivic homotopy theory is related but different. I hadn’t seen the constructions at motive though, they should probably also be moved to mixed motive.
I created symmetric sequence and added a general definition at spectrum, following Ayoub’s thesis.
Next is monoidal and model structures on spectra.
edit: there is some stuff on the symmetric monoidal structures now.
I added today’s Caramello’s paper on motives from the topos point of view into motive entry. I also merged the references section with lost subsection a page above also containing and named as “references”. That was somehow lost in organization of the page. The merger may need further microorganization (the moved references are still under “general”, only one reference was double listed).
I have only glanced at Olivia’s paper. Is my impression correct that this motivation is different from the one stated in cohomology#ToposTheory ?
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