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Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make each one of these constructions fail to be the real category of pure and mixed motives.

In any case, there are a lot of constructions of what is supposed to be the category of pure and mixed motives. In the following, I will cite some of these ones. Please, feel free to edit, comment and add additional models for the category of motives.

I will focus on motives over a field $k$ or over a base scheme $S$. $\Lambda$ and $F$ will denote the coefficient rings. Furthermore $B$, $C$, $D$ will denote the Standard Conjectures.

Pure Motives

For the pure case, there are 3 main constructions when $F$ is of characteristic zero:

(Ch) Numerical (and homological) Chow motives $M_{num} (k, F)$ (resp., $M_{hom} (k, F)$);

(AH) Absolute Hodge cycles $M_{AH} (k, F)$ for $k$ of characteristic $0$;

(Mot) Motivated cycles $M_{mot} (k, F)$ for $k$ of characteristic $0$.

As I understand, what's missing is the following:

(Ch) Abelian semisimple, but it needs $D$ to have the universal property for Weil cohomologies and $C$ to be neutral Tannakian;

(AH) neutral Tannakian, but it needs the Hodge conjecture to satisfy the universal property for Weil cohomologies and $D$ to be abelian semisimple ;

(Mot) neutral Tannakian, but it needs $B$ to satisfy the universal property for Weil cohomologies and $D$ to be abelian semisimple.

1) Are the problems cited above the only ones to prevent any of these categories to be the real category of pure motives? Furthermore are there construction for $F$ of positive characteristic?

Mixed Motives

For the mixed case, there are $6$ main constructions (where only the first one is the abelian category of mixed motives, whereas the others are only the derived category):

(MR) Mixed motives for absolute Hodge cycles $MM_{AH} (k, F)$ when $k$ is of characteristic $0$ and $F$ is of characteristic $0$, i.e., the subcategory of compatible mixed realizations generated by the image of the realization of smooth quasi-projective varieties over $k$.

(V) Voevodsky motives $DM(S, \Lambda)$ for $S$ Noetherian;

(etV) Étale Voevodsky motives $DM_{ét} (S, \Lambda)$ for $S$ Noetherian;

(L) Levine motives $\mathcal{DM} (k, \Lambda)$ for $k$ a perfect field;

(H) Hanamura motives $M\mathcal{M} (k, \Lambda)$;

(N) Nori motives $\mathcal{MM}_{Nori} (k, \Lambda)$ for $k$ embeddable in $\mathbb{C}$.

As I understand, what's missing is the following:

(MR) Neutral Tannakian, but contains the same problems as (AH) for its subcategory of pure motives;

(V) It doesn't have a heart for $\Lambda = \mathbb{Z}$ and the Betti realization is not conservative;

(etV) It may have a heart for $\Lambda = \mathbb{Z}$, but it's not known;

(L) It coincides with (V) for $k$ of characteristic zero and for $k$ perfect and $F = \mathbb{Q}$. I don't know what happens in integral coefficients;

(H) I know nothing about it;

(N) It's compatible with the motivic t-structure of (V) for $\Lambda$ a Dedekind domain or a field and $k$ embeddable in $\mathbb{C}$ and, hence, it has the same problems as (V);

2) Are the problems cited above the only ones to prevent any of these categories to be the real category of mixed motives? Furthermore what are all the known comparisons between all these constructions beside the ones already cited above?

3) What happens in mixed characteristic with equal characteristic residue fields? How does one define the motivic t-structure when there is no Betti realization?

Thanks in advance.

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    $\begingroup$ Aren't you askiing too much for a single question? Note in particular that is not really clear what should be the "real category of motives" if one considers integral coefficients. So, I will just tell you that I have proved that Hanamura's motives are anti-isomorphic to Voevodsky's ones (in characteristic 0, but this assumption can be dropped thanks to results of S. Kelly); note that Hanamura's motives are $\mathbb{Q}$-linear by definition (yet a certain integral version of this category exists also).:) $\endgroup$ – Mikhail Bondarko Dec 2 '16 at 21:46
  • $\begingroup$ @MikhailBondarko Yes, I agree I'm asking too much. I'm actually just trying to compile all comparisons and failures in one place. I think this would help other people too since most of the results are scattered around a lot of papers. But about your comment, where can I find about this integral version of Hanamura motives? And where can I find a precise statement of this comparison? $\endgroup$ – user40276 Dec 2 '16 at 21:54
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    $\begingroup$ Just read my paper: arxiv.org/pdf/math/0601713v9.pdf Also, though I would prefer you to say what "real motives" are, I would like to say that the category MR is definitely "wrong". $\endgroup$ – Mikhail Bondarko Dec 2 '16 at 21:56
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    $\begingroup$ Two more commens.1) :For motives over a "general" bases one should define the motivic t-structure in terms of perverse etale (co)homology. 2) (MR) does not satisfy any "reasonable" universality property. $\endgroup$ – Mikhail Bondarko Dec 3 '16 at 19:56
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    $\begingroup$ Alexander Beilinson proved that the standard conjectures would follow from the existence of a motivic $t$-structure: arxiv.org/abs/1006.1116 $\endgroup$ – TKe Dec 4 '16 at 7:04

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