Do notions of algebraic cycles and motives extend to Lurie's derived and/or spectral schemes? Would one expect to construct them more easily in these more "flexible" settings, and then carry them over to ordinary schemes with $\pi_0$?

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Do notions of algebraic cycles and motives extend to Lurie's derived and/or spectral schemes? Would one expect to construct them more easily in these more "flexible" settings, and then carry them over to ordinary schemes with $\pi_0$?

everymap of ring spectra is a quotient map. You can do more sophisticated things but on the whole there's no theory I am aware of (cont.). $\endgroup$ – Denis Nardin Apr 3 at 7:38