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have been editing a bit more at topological cyclic homology, expanded the Idea-section a bit more, added and reorganized references, added some more cross-links.
I included the recent reference
Thanks. We should also link to this from Frobenius morphism. They sort of give the concept a deeper home in stable homotopy theory.
Right. I saw you mentioned Frobenius morphisms on g+. You consider that the main advance of the paper, or one of them? That’s presumably what’s described as
We start in Section IV.1 by defining a certain Frobenius-type map $R \to R^{t C_p}$ defined on any $\mathbb{E}_{\infty}$-ring spectrum $R$, which in the case of classical rings recovers the usual Frobenius on $\pi_0$, and the Steenrod operations on higher homotopy groups.
So how are we to understand their claim of “using only homotopy-invariant notions”? If this is a more synthetic approach, can it be captured by some HoTT approach? It’s true there’s plenty about homotopy orbits and fixed points, so dependent sums and products, but I wonder how quickly we need something more specific. E.g., is that norm map $Nm_G : X_{h G} \to X^{h G}$ only available for finite groups acting on spectra?
As for that last point, Definition I.1.10 spells out broader conditions.
I saw you mentioned Frobenius morphisms on g+. You consider that the main advance of the paper, or one of them?
The main theorems of the article improve on ideas that had been known to some extent before. The statement about the Frobenius morphism struck me as a remarkable new insight, connecting number theory with stable homotopy theory. According to Thomas, Peter said about this that he wouldn’t have thought that there was still something new for him to learn about Frobenius.
I am glad that the article is finally out, so that I can read up on the details (having heard various talks and explanations before). I do suspect this is closely related to the brane bouquet story, since after all there double dimensional reduction is found to be implemented by passing to cyclic cohomology, and topological cyclic (co-)homology is just the proper $\infty$-version of that. Also the way that cyclotomic spectra encode $S^1$-equivariant structure in terms of fixed points of underlying $\mathbb{Z}/p\mathbb{Z} \subset S^1$-actions is possibly related to the A-type ADE singularities in this business, which involve exactly this: $\mathbb{Z}/\mathbb{Z}p$-actions inside $S^1$ on the $S^1$-circle fibers of the M-theory fibration. Presently I don’t see further than this coincidence, but I suspect there is more to it.
I have been adding pointers to relevant parts of Nikolaus-Scholze 17 to Tate spectrum, cyclotomic spectrum, topological cyclic homology, norm map, created a minimum at Farrell-Tate cohomology and added the plain definition for $E_\infty$-rings to Frobenius morphism (here)
Not done yet, but need to quit for tonight.
I added a little more at Farrell-Tate cohomology, including the extra condition on Farrell’s generalisation
What is called Farrell-Tate cohomology is a generalization of this construction to possibly infinite discrete groups $G$ of finite virtual cohomological dimension.
We have cohomological dimension, so I’ve tried to find out about the virtual variety. I found a couple of attempts to define this, now at virtual cohomological dimension. Not sure how to piece them together.
Recovered via the WaybackMachine the pdf link for this item:
To me the change in https://ncatlab.org/nlab/revision/diff/topological+cyclic+homology/23 seems contraproductive, it seems it was to correct the English and the sentence now parses worse, to me at least.
Revisions also seem somewhat worse in https://ncatlab.org/nlab/revision/diff/strict+2-category/50 and https://ncatlab.org/nlab/revision/diff/simple+object/25.
Undid the funny edit from revision 23 by “Rutherford”
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