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In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the setting of (infinity, 1)-categories, the stable (infinity, 1)-category Spt of spectra is the analogue of Ab. Hence in derived algebraic geometry one studied spaces locally modelled on E-infinity rings, i.e. commutative algebra objects in Spt.

Spt is by definition the stabilization of the category of spaces. In motivic homotopy theory (Morel-Voevodsky), one considers motivic spaces (A^1-homotopy invariant Nisnevich sheaves on the category of smooth k-schemes, say) and their stabilization MSpt (motivic spectra). Hence one could probably consider what might be called "motivic derived algebraic geometry", where spaces are locally modelled on commutative algebra objects in MSpt.

(Then one could consider derived motivic spaces, derived motivic spectra, and motivic 2-derived algebraic geometry, etc...)

I apologize for the rather ridiculous question, but: could this possibly result in something useful? Maybe another way to put the question is: how do we know that E-infinity rings give us the most complete picture? Or perhaps there are technical reasons that prevent one from using other categories than Spt?

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I have no idea what I'm talking about... but if such a thing existed where your generalized ring objects lived in motivic spectra, then perhaps you could build a motivic tmf a la Lurie? –  Dylan Wilson Feb 9 at 22:56
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I guess everything depends on what your goal is. Derived or homotopical algebraic geometry started with the purpose of building a framework to prove some theorems, rather than a per se generalization of classical stuff. –  Fernando Muro Feb 9 at 22:56
    
@FernandoMuro, right this is what I'm wondering about: whether such a construction could have any interesting applications for which E-infinity derived geometry would be insufficient. –  Adeel Feb 10 at 11:12
    
@DylanWilson, that sounds pretty interesting, though I know nothing about tmf. –  Adeel Feb 10 at 11:13
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I believe this is a sensible construction, since there are numerous objects in algebraic geometry (algebraic K-theory, algebraic cobordism etc) that seem to be formal counterparts of similar objects in classical homotopy theory, which respectively are often studied via algebraic geometry. However, such objects are very big and complicated sheaves, so I doubt that there you would be able to prove any complex geometric statements about them, except for some general abstract nonsense and calculations at specific points (which may be still very non-trivial and interesting). –  Anton Fetisov Feb 10 at 21:54

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