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?

noidea 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 22:56per segeneralization of classical stuff. – Fernando Muro Feb 9 at 22:56