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As a physicist who has read almost nothing on the beautiful theories of motives and motivic cohomology, I have some philosophical questions on motivic cohomology. I apologize first for my naive understanding, which may trivialize my question in the first place. I find the following MO post is very illuminating.

What is the relationship between motivic cohomology and the theory of motives?

The conjectured abelian category of mixed motives, which is still shrouded in mystery, is expected to be the universal cohomology theory of varieties. But from the literature in number theory, it is the motivic cohomology that appears more frequently, e.g. it plays a central role in Beilinson's conjecture on values of $L$-function. It gives me the impression (probably wrong impression) that motivic cohomology seems to be in some sense more fundamental in number theory. Anyone who can say more on this (and correct me)?

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  • $\begingroup$ It is expected that motivic cohomology is given by some Ext group in the (still non-existent) category of mixed motives. We don't know yet the existence of this category, but by Voevodsky et al. we can construct what should be its derived category, which is a triangulated category. So you can define motivic cohomology as some Hom group in this triangulated category. $\endgroup$
    – François Brunault
    Commented Mar 26, 2018 at 22:20
  • $\begingroup$ Also, I think "The category of mixed motives is expected to be the universal cohomology theory of varieties" is a bit misleading. Maybe you mean that there should be a functor from varieties to mixed motives such that all the "classical" cohomology theories factor through this functor. $\endgroup$
    – François Brunault
    Commented Mar 26, 2018 at 22:40

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