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I am not very familiar with motif theory, but I do know a little about Hodge theory. I view (mixed) motif theory as an enhancement of (mixed) Hodge structures.

Q1. Is (mixed) motivic sheaf theory an enhancement of the theory of (mixed) Hodge modules?

A mixed Hodge module is a kind of "constructible sheaf valued in mixed Hodge structures". Of course, this is a very naive and incorrect assertion. In fact, Saito's theory of mixed Hodge Module is more complex than "sheaves valued in mixed Hodge structures". So, I ask a second question.

Q2. Is a (mixed) motivic sheaf a "sheaf valued in (mixed) motives"? Does the complexity of the theory of (mixed) motivic sheaves have similar origins as the complexity of the theory of (mixed) Hodge modules?

Sorry for the vagueness of these questions. Also, any intuitive explanation of motivic sheaves (without answering my questions) are welcome.

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    $\begingroup$ Beware that 'motives' (or 'motifs') mean a lot of different things depending on whom you ask (e.g. Chow motives, Nori motives, André motives, Voevodsky motives, ...). A more precise version of your first two sentences is that motives are a (partially conjectural) enhancement of Hodge structures, and mixed motives a (partially conjectural) enhancement of mixed Hodge structures. (But many people working in motivic homotopy theory drop the word 'mixed', which is fair enough if that's the only thing you're studying.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 19, 2022 at 8:16
  • $\begingroup$ @R. van Dobben de Bruyn Thanks! I've edited my question. $\endgroup$
    – D.Namrebod
    Commented May 20, 2022 at 2:08

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Q1. At least partially yes.:) There should exist a nice system of connecting functors. This text https://arxiv.org/abs/1801.10129 should give much of this theory; I don't know why it is not published yet.

Q2. I would say "no". The problem is that is really difficult to define mixed motives (even over an algebraically closed field). We only have some recipes for defining the corresponding triangulated categories, and they "do not depend on the choice of the base scheme" much.

Note that the original idea of Grothendieck is that motives should be "defined in terms of varieties". And this is more or less the case for triangulated motivic categories. As far as I know, that difference with Hodge modules is that one defines all motivic categories without using vanishing sheaf functors (however, these can be defined eventually).

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  • $\begingroup$ Thanks for the very helpful answer! I couldn't understand the sentence "they "do not depend on the choice of the base scheme" much". You mean, in the construction of the category of motivic sheaves, you do not treat motivic sheaves as sheaves? $\endgroup$
    – D.Namrebod
    Commented May 23, 2022 at 8:55
  • $\begingroup$ Well, you usually start from complexes of sheaves (or presheaves) on the site of all schemes that are smooth over the base. After certain operations you obtain a candidate for the derived category of motives over this base. However, it (currently?) does not make much sense to look for "single" mixed motivic sheaves inside this category, and to treat those as sheaves. $\endgroup$
    – Mikhail Bondarko
    Commented May 24, 2022 at 10:55
  • $\begingroup$ I see. Thanks for the clarification! $\endgroup$
    – D.Namrebod
    Commented May 25, 2022 at 2:49

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