# different “derived structure” in derived algebraic geometry

In derived algebraic geometry there are several different setting,i.e., sometimes we use $E_{\infty}$ring,somethings we use dg-algebra,... It is for different situations. But could someone give some examples illustrating under what problem we use relevant "derived structure" ($E_{\infty}$ ring, dg-algebra...)? Some motivations?

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In mirror symmetry one of the things we are interested in is deformations of the relevant dg/A-infty algebras/categories. The relevant deformation spaces are, or should be, related to, for instance, quantum cohomology (see Barannikov-Kontsevich or Kontsevich's original homological mirror symmetry paper). As for E-infty rings, see Lurie's survey on elliptic cohomology for some motivation. –  Kevin H. Lin Jul 26 '10 at 8:53
At least in the context of Toen-Vezzosi's HAG (vol 2, if I remember correctly), derived algebraic geometry is specifically simplicially-enriched algebraic geometry, while they use "complicial algebraic geometry" and "brave new algebraic geometry" for enrichment in complicial sets and spectra respectively. –  Harry Gindi Jul 26 '10 at 8:58
A frequent remark is that dgas are only good in characteristic zero, while in positive characteristic one has to use simplicial rings or E_k or E_\infty rings. See math.harvard.edu/~lurie/papers/moduli.pdf for a case, where this thought is worked out. –  Lennart Meier Jul 26 '10 at 12:46

I minor point is that complex orientations are not necessarily represented by $E_{\infty}$ ring spectra. Having an $E_{\infty}$ complex orientation is something one has to work for. See recent work of Niles Johnson and Justin Noel or the thesis of Matt Ando or Jim McClure. –  Sean Tilson Oct 17 '12 at 21:56