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started curved dg-algebra
One of the typical sources of curved dg-algebras is any geometric source of a usual dg algebra, but with a gerbe present. So “curved” is like “gerbal”. How about the categorification. What would be the case corresponding to a 2-gerbe ? In other words, how to twist the definition of a dg algebra (or even a dg category) in the presence of a 2-gerbe ?
Of course, I ask this question being happy after hearing the talk by Jonathan Block in Cardiff; his paper from Bott’s volume is added at curved dg-algebra. In that paper he replaces category of modules over a curved dga with a different (usual, non-curved) dg-category. The latter comes up naturally in a number of situations as the correct higher category (of coherent sheaves and alike) for certain geometrical situations.
his paper from Bott’s volume is added at curved dg-algebra. In that paper he replaces category of modules over a curved dga with a different (usual, non-curved) dg-category. The latter comes up naturally in a number of situations as the correct higher category (of coherent sheaves and alike) for certain geometrical situations.
I suppose what you are alluding to is what Block describes in more detail in section 2 of Mukai duality for gerbes with connection (arXiv:0803.1529v2).
I need to refresh my memory on this.
So you were at the same conference as Danny was, I suppose? Do you have a link?
By the way, Zoran: thanks for writing this. I realize that I was missing your input here!
The paper you quote has more emphasis on operator algebraic version, where dg-categories are in the sense of complexis in bornological etc. setup. I will look more on the details later. Now, yes, the curved dga appears the way as in section 2. The paper I quote has a main point of replacing category of modules with another dg-category which is more appropriate from geometrical/homological point of view.
There was a good turn out at Cardiff. I was down there as were Danny and Zoran. We had some excellent talks, (see here.
I was very impressed by Jonathan Block’s talk as it seemed to indicate a real sense of a strong link between the Connes view of NCG and the dg-category one, yet needed higher categories to make sense of it.
(see here.
Ah, thanks!.
In Connes’ talk, was there an announcement of anything new, or was it just a review?
It was a public lecture, so mostly a review. It followed on from a talk the previous evening on the LHC and the mass of the Higgs from the physicist in charge of the LHC (who happens to be from Wales). :-) The Connes work gives the ‘wrong’ mass so he did discuss what sort of changes might be needed to correct the model.
I see. Did he comment on Block’s work? In question session or privately?
Not to my knowledge.
Connes did not attend Block’s talk as that one was on Friday and Connes left before Thursday session.
Kontsevich has mentioned in his public lectures on noncommutative motives and alike topics (when he talks about aspects related to cyclic homology in that setup) that he expected that if among the categories enriched over chain complexes, or more generally, A-infinity, or even more properly categories enriched over spectra, one singles good ones he considers (words like smooth proper separated etc.) than there is probably a way to get some analytic/operator algebra counterpart; he mentioned using something like nuclear spaces in a possible formalism. But he did not realize such a correspondence so far (it could be like some sort of noncommutative and categorified GAGA).
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