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Started a bare minimum at cyclotomic spectrum. So far it’s essentially just a pointer to the canonical reference by Blumberg-Mandell. (Thomas Nikolaus and Peter Scholze have a new foundation of the theory in preparation for which notes however are not public yet, also Clark Barwick has something in preparation, for which you may find notes by looking at his website and being clever in deducing hidden URLs, he says.)
For the moment the only fact that I have actually recorded in the entry is a fact that is trivial for anyone familiar with the theory,but which looks interesting from the point of view of the story at Generalized cohomology of M2/M5-branes (schreiber): the global equivariant sphere spectrum for all the cyclic groups (all the A-type finite groups in the ADE classification…) carries canonical cyclotomic structure and as such is the tensor unit among cyclotomic spectra.
Apart from mentioning this, I have added brief cross-links with topological cyclic homology, equivariant sphere spectrum, cyclic group and maybe other entries.
hidden URLs
So you take one linked to like Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin and change the number? Seems to work for ’2’.
There’s a brief definition of cyclonic spectra in Glasman’s research statement
cyclonic spectra: spectra with $S^1$-action that carry $G$-fixed point data for finite subgroups $G \leq S^1$ but no $S^1$-fixed point data.
These homotopy theorists seem to have grand plans. I wonder what
[BDG+15] Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah. Equivariant higher categories and equivariant higher algebra. In progress, 2015.
will bring.
I have added a further sentence-and-a-half on the idea of cyclotomic structure: For $A$ a connective $E_\infty$-ring then $THH(A)$ is the $E_\infty$-ring of functions on the free loop space of $Spec(A)$ and cyclotomic structure reflects the following structure of free loop spaces: loops that repeat with perdiod $p$ (hence $C_p$-fixedpoints in the space of loops) are equivalent to plain loops.
Also added the statement that the cyclotomic sphere spectrum is $\mathbb{S}$ regarded as $THH(\mathbb{S}) \simeq \mathbb{S}$.
I have added the Nikolaus-Scholze definition here and wanted to start writing out examples here, but nothing really yet.
The construction of the cyclotomic structure on the sphere spectrum in Nikolaus-Scholze 17, example II.1.2 (ii), does that depend in any way on the spectrum being the sphere spectrum, or is it a general statement for spectra equipped with trivial circle action?
By the first lines on p. 31, this boils down to asking what it is about the sphere spectrum that makes the equivalence
$\mathbb{S}^{S^1} \simeq \left( \mathbb{S}^{C_p}\right)^{S^1/C_p}$work. I would have thought this is complely general, being the decomposition of the right base change along $B S^1 \to \ast$ into base change along the composite
$B S^1 \overset{}{\longrightarrow} B (S^1/C_p) \longrightarrow \ast \,.$But maybe I am missing something.
Is it really the case, as claimed on the page, that the $S^1$-equivariant sphere spectrum of Blumberg-Mandell 13, example 4.9, is the same as the sphere spectrum with trivial circle action?
Notice that we are going now across the equivalence between “genuine” cyclotomic spectra, underlying which are genuine $S^1$-equivariant spectra, and Nikolaus-Scholze cyclotomic spectra, underlying which is just a plain spectrum equipped with a plain $S^1$-action (hence not a genuine but a “doubly naive” $S^1$-equivariant structure).
The claim of that example II.1.2 (ii) is that to get the cyclotomic sphere in the second picture, one is to consider the trivial $S^1$-action. This does not mean that the genuine $S^1$-equivariant sphere spectrum (which is a very different object altogether) is trivial in its $S^1$-dependence.
What do you mean “we are going now across the equivalence”? The page cyclotomic spectrum has
The tensor unit in the symmetric monoidal (infinity,1)-category of cyclotomic spectra is the cyclotomic sphere spectrum from example 3.2 (Blumberg-Mandell 13, example 4.9)
So example 3.2 is the Nikolaus-Scholze (second picture) spectrum and example 4.9 of Blumberg-Mandell 13 is a (first picture) genuine spectrum:
Example 4.9. The $S^1$-equivariant sphere spectrum has a canonical structure as a cyclotomic spectrum induced by the canonical isomorphisms…
So example 3.2 is the Nikolaus-Scholze (second picture) spectrum and example 4.9 of Blumberg-Mandell 13 is a (first picture) genuine spectrum:
Yes, they are both desribing the cyclotomic sphere, but on the two sides of the equivalence
$CycSpec_{-}^{gen} \simeq CycSpec_-$(where the subscript is meant to be “bounded below”).
I thought you were asking if it can be right that the underlying $S^1$-action on the cyclotomic sphere spectrum is trivial, and I took that as wondering whether it could be true that the genuine $S^1$-equivariant sphere spectrum somehow has trival $S^1$-dependence. I tried to say that the first is true without implying the latter, because these $S^1$-structures on the sphere spectrum are on the two different sides of the above equivalence, and are not “the same”.
Ah, so that’s
Theorem II.6.9.The functor $CycSp^{gen} \to CycSp$ induces an equivalence between the subcategories of those objects whose underlying non-equivariant spectra are bounded below.
Perhaps that could be clearer on the page, so I’ve added
More formally, theorem II.6.9 states that the forgetful functor $CycSp^{gen} \to CycSp$ (Prop. II.3.4) induces an equivalence between the subcategories of those objects whose underlying non-equivariant spectra are bounded below.
Okay, thanks!
I have now checked with Thomas. Indeed there is nothing special about the sphere in the example II.1.2(ii). They come back to this in the middle of p. 126, where they consider the construction generally, and call it $(-)^{triv} : Spectra \to CycSpectra$.
I have edited the Examples-section accordingly here.
Also, in the Definition section here I added the remark that those structure maps $F_p \colon X \to X^{t C_p}$ of a cyclotomic spectrum are “the Frobenius morphisms” .
I suppose this gives a neat realization of the perspective on Frobenius morphisms as arithmetic translation operators as in the section Motivation at Borger’s absolute geometry.
Since we’re seeing $p$-completions of cyclotomic spectra, III.1.7: $\Delta_p: X \to (X \otimes \cdots \otimes X)^{t C_p}$, how about other parts of the cohomology hexagon, $p$-localization, etc.?
I wonder whether the category of these spectra is the tangent $(\infty, 1)$-category of an interesting category at $\ast$. I see from here
A cyclotomic spectrum can be built by taking the suspension spectrum of a cyclotomic space
and
Def 3.6: A cyclotomic space $A$ is an $S^1$-equivariant space together with compatible equivalences…
Now what kind of category would these spaces form?
Thanks for the pointer. That’s a good thing to record. I made a start at cyclotomic space.
Hm, now I’d need go checking: do we sometimes ask for those equivalence $X^{C_p} \to X$ only at primes $p$ and sometimes at all natural numbers?
(Need to run. Back later.)
The references I’ve seen say all natural numbers. Also we need to speak of the compatibility of maps.
So I’ve integrated what I found in
I’m sure it can be written more nicely.
The references I’ve seen say all natural numbers.
In Nikolaus-Scholze they use just the prime numbers, no? Maybe it does not matter due to the primary decomposition of cyclic groups. But that does not quite reduce to $C_p$-s just to $C_{p^n}$-s. Hm.
Is this maybe related to the difference between their version and genuine cyclotomic spectra? When they come to define the latter, there’s something like those equivalences for all $n$ in Definition II.3.6 on p. 46, and footnote 21
Their commutative diagram in [BM15, Definition 4.7] looks different from ours, and does not seem to ask for a relation between $\Phi_{m n}$ and $\Phi_m$, $\Phi_n$; we believe ours is the correct one, following [HM97, Definition 2.2].
By the way, how are we to think of their version? Is it that we really want the genuine form, but are to find as more convenient their version for spectra bounded below, as shown to be equivalent?
By the way that [BM15, Definition 4.7] they refer to, takes the term for $m n$, and then uses $m n = n m$ to provide two equivalences.
p. 6
Note that contrary to the case of orthogonal cyclotomic spectra, we do not ask for any compatibility between the maps $\phi_p$ for different primes $p$.
New today:
Title: Comparing cyclotomic structures on different models for topological Hochschild homology
Authors: Emanuele Dotto, Cary Malkiewich, Irakli Patchkoria, Steffen Sagave, Calvin Woo
The topological Hochschild homology $THH(A)$ of an orthogonal ring spectrum $A$ can be defined by evaluating the cyclic bar construction on $A$ or by applying B"okstedt’s original definition of $THH$ to $A$. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for $THH(A)$. This implies that the two versions of topological cyclic homology resulting from these variants of $THH(A)$ are equivalent. \ ( https://arxiv.org/abs/1707.07862, 33kb)
added minimal cross-link with cycle category and cyclic set and cyclic object
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