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have created an entry for the new book: Higher Algebra
It is interesting, he used DAG i-iv and vi and excluded v, added material and changed title. So DAG-v Structured spaces will be part of yet another volume.
and changed title
I am glad he finally did! That “derived algebraic geometry” over the whole thing tricked people into thinking about it in the wrong way. In particular into thinking that it does not concern them.
and excluded v
Seems to make sense to. That is the article that is genuinely about geometry. His next book should be titled Higher Geometry . That would complete the Isbell duality Dreiklang :
Topos theory to rule them all, and then in there Isbell duality between algebra and geometry.
That "derived algebraic geometry" over the whole thing tricked people into thinking about it in the wrong way.
This depends on person's background, I have a bit different feeling. In many of your writings it confused me for months that you call the cosheaves of algebras "quantity" by some Lawvere's terminology. For most of us in noncommutative and algebraic geometry we talk about the same duality under space-algebra of functions. One may criticise that it is not always an algebra, but one can equally criticise that it is nothing about quantity (for me, with physics terminological background, a quantity would be a global section at best, by no means a cosheaf). Of course, algebra generalizes to other structures living over space and defining it, including topoi and higher topoi. So in this vein, when one dismissed the space in topological sense and replaces it by thge structures over it, including the topoi of infinity sheaves, I find a good terminology derived algebraic geometry. For example, if one looks at manifolds by gluing open subsets this is not that algebraic approach as when one goes to look at probes and coprobes and hence gets lead to the generalized algebras of observables. This is in a way algebraic approach to differential geometry, or a generalized algebraic formulation. I understand that not everybody will see from that perspective but it is not any less true than with "quantity". I think that Lurie's title DAG does lead me to the correct way of thinking on its scope. If you read Toen-Vezzosi work on Segal topoi most of it is in almost the same level of generality (replace infinity categories with model categories at places).
On the other hand, I like your quoting Isbell's duality again...
that usage of “quantity” I haven’t really followed up much, it’s certainly very non-standard.
But I don’t think that’s what I mean. What I mean is that the title “derived algebraic geometry” didn’t properly indicate (at least not how it was usually taken) that
a) the developments were fully general abstract higher category theory that was related to applications in geometry only in so far as any piece of category theory is related to everything else;
b) in as far as the developments were about geometry proper, they were fully general geometry far more general than what is usually understood as “algebraic geometry”, derived or not.
I recently had a chat with some professors over this. They recounted some meeting where people discussed how to name these things. Somebody boldly and proudly suggested “You know, we should just drop the “derived” and just call this “algebraic geometry”.”
I think, no, we should also discard the “algebraic” and just call this “geometry”. And dually algebra.
I think one reason why not more people who think of themselves as category theirsist got into these developments is because they think it’s about algebraic geometry, not about category theory.
I think, no, we should also discard the "algebraic" and just call this "geometry".
Surely, unless you talk to the traditional geometers. Today I gave a lecture in topology seminar, and a bit of introducing stiff like Kan extensions and thier homotopy for geometric topologists was clearly too much of abstracies which they do not perceive as being into topology. The natural generality of work is accross boundaries and unless you are in supermodern center it is a problem. One could say optimistically just a problem of generations, but I am not sure that in 20 years this problem will be gone for today's notions, though some shift will happen. Some people still do geoemtry via axiomatic systems.
Higher Algebra includes
Remark 2.0.0.8. An alternate approach to the theory of -operads has been proposed by Cisinski and Moerdijk, based on the formalism of dendroidal sets (see [29] and [30]). It seems overwhelmingly likely that their theory is equivalent to the one presented in this chapter. More precisely, there should be a Quillen equivalence between the category of dendroidal sets and the category of -preoperads which we describe in paragraph 2.4.
Yes, people in utrecht are working on it. You can see that at (infinity,1)-operad there is still a sketch of a proof of mine of a Quillen adjunction between the Cisinski-Moerdijk-Weiss definition and Lurie’s, only that I never tried to patch that last step.
Gijs Heuts has a lot of ideas at least about related things. He might get close before he finishes his MSc, or maybe afterwards. he’ll go to Harvard. At a rough level it seems pretty clear what one needs to do, but some details may seem a little intricate.
What does this mean, Urs?
Topos theory to rule them all, and then in there Isbell duality between algebra and geometry.
I thought topos theory was seen as being on the geometric side. E.g., from Higher Geometry,
Higher geometry is typically built on -topos theory.
A sudden thought, is there a higher version of Freyd’s AT-category result?
today I have been adding a dozen or so further linked keywords to Higher Algebra and created the corresponding entries. Mostly just basic definitions from the book. I needed to collect some material…
Just wanted to point out that there is a new version of Higher Algebra available on Lurie’s website now: pdf.
Updated with some additional material on duality in monoidal infty-categories and Koszul duality for E_k algebras.
Thanks, I would have missed this. But so I suppose the links at Higher Algebra are automatically up-to-date, right?
The name of the pdf had changed from HigherAlgebra.pdf to higheralgebra.pdf for some reason, so I corrected that.
Just to say that there is a new version of Lurie’s HA with, it seems, a lot of new material . The link in the nLab page will take you to it.
Last update: September 2017; rewrote section on the associative operad and added material on A_n algebras.
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