In the paper "Motivic model categories and motivic derived algebraic geometry", Yuki Kato, whose email-address I sadly couldn't find out, describes a procedure to "motivy" the objects of any $(\infty,1)$-category by doing the construction used to get motivic spaces over an arbitrary base. He then applies this construction to $(\infty,1)$-Cat itself to get motivic categories themselves. The abstract reads "This result gives us the motivic versions of the formulations of spectral schemes and spectral Deligne--Mumford stacks introduced by Lurie."

Is this something algebraic geometers are after, and if so why? My knowledge of the role motivic spaces play is only vage, but I mainly understand them as a means to get to the derived category of motives, which is the "universal category of (nice) cohomology coefficients" in some way. Why do they make good building blocks for generalized schemes? What new properties would the resulting spaces be expected to have? What differentiates the motivic versions of their categories from their ordinary ones?

I realize that the answer to this question might possibly known to only one person. If so, I apologize for wasting your time.

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    $\begingroup$ There is definitely some value in having "motivic categories" of some sort, because those are natural targets for realization functors. I haven't read the paper though, and I think that you'll need some additional hypotheses on your categories to make it worthwhile. For a similar idea in the setting of equivariant homotopy theory you can see this paper. $\endgroup$ – Denis Nardin Oct 31 '17 at 20:06
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    $\begingroup$ Kato's email address can be found at the end of his preprint. $\endgroup$ – j.c. Oct 31 '17 at 22:37
  • $\begingroup$ Ah, thank you! I tried to google him, but nothing came up. I will ask him direktly then. $\endgroup$ – Alexander Praehauser Oct 31 '17 at 23:17

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