It appears to me (though I may be wrong) that the common opinion is that the main difference between two is that Rosenberg's version of noncommutative algebraic geometry mainly concerns as noncommutative spaces represented by abelian categories while derived noncommutative geometry as Kontsevich, Orlov, Efimov, Toen etc. study more general spaces represented by enhanced triangulated categories (DG-categories, $A_{\infty}$-categories, stable $\infty$-categories etc.)

However, it seems that it is not the case: Rosenberg introduced his notion of a (left) spectrum not only to abelian categories, but also to triangulated categories (this matching derived geometry in the level of generality) and to right exact categories with weak equivalences (which, apparently, is a generalization of Barwick's notion of an exact $\infty$-category where "exactness" is one-sided) (see here and here).

Apparently, the key difference is the notion of a spectrum of an underlying noncommutative space: somehow, derived noncommutative geometers make do without it. It appears that the prescence of spectrum makes the idea of the category being the "category of quasicoherent sheaves on a noncommutative space" more rigorous in a categorical since then a noncommutative space is represented by a categorical analogue of a locally ringed space, namely a stack of certain categories.

I would like to know more about differences and similarities between this two approaches. In particular

What the spectrum does in Rosenberg's noncommutative geometry and how Kontsevich's derived geometry makes do without it? Would it be possible to introduce the notion of spectrum into derived noncommutative geometry?


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