It appears to me (though I may be wrong) that the common opinion is that the main difference between derived noncommutative geometry and Rosenberg's noncommutative 'spaces' is that Rosenberg's version of noncommutative algebraic geometry mainly concerns noncommutative spaces represented by abelian categories while derived noncommutative geometry as Kontsevich, Orlov, Efimov, Toën 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 for abelian categories, but also for 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 presence of a spectrum makes the idea of the category being the "category of quasicoherent sheaves on a noncommutative space" more rigorous in a categorical sense 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 these two approaches. In particular

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

  • $\begingroup$ Your links to Rosenberg's articles on triangulated categories are no longer accessible, and I wonder to what extent this could be used to reconstruct the original scheme in the commutative case? $\endgroup$
    – Z. M
    Jul 3, 2022 at 16:50
  • $\begingroup$ @Z.M The two sources are available on the MPIM preprint server here and here but I can't quite find the corresponding statements in them. (I have submitted an edit to update the links.) I think that to recover a variety in the commutative setting would generally require more than the triangulated structure (e.g. the Balmer spectrum of $\mathrm D^{\mathrm{perf}} (X)$ with its tensor triangulated structure given by the derived tensor product $\otimes_{\mathcal O_X}^{\mathbf{L}}$ can recover $X$). $\endgroup$ Jul 3, 2022 at 20:52
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    $\begingroup$ @SeverinBarmeier I gave them a glance, but I do not see anywhere he envisaged to apply such a theory to derived categories. Weak equivalences are introduced, yes, but I don't see anywhere weak equivalences are given by quasi-isomorphisms. I am aware of Tanakian reconstruction, but it is a different story. I want to understand how much information is there noncommutatively. $\endgroup$
    – Z. M
    Jul 3, 2022 at 21:31


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