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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2015
    • (edited Dec 15th 2015)

    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.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 15th 2015

    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 1S^1-action that carry GG-fixed point data for finite subgroups GS 1G \leq S^1 but no S 1S^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.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2015
    • (edited Dec 16th 2015)

    I have added a further sentence-and-a-half on the idea of cyclotomic structure: For AA a connective E E_\infty-ring then THH(A)THH(A) is the E E_\infty-ring of functions on the free loop space of Spec(A)Spec(A) and cyclotomic structure reflects the following structure of free loop spaces: loops that repeat with perdiod pp (hence C pC_p-fixedpoints in the space of loops) are equivalent to plain loops.

    Also added the statement that the cyclotomic sphere spectrum is 𝕊\mathbb{S} regarded as THH(𝕊)𝕊THH(\mathbb{S}) \simeq \mathbb{S}.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2017
    • (edited Jul 24th 2017)

    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

    𝕊 S 1(𝕊 C p) S 1/C p \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 BS 1*B S^1 \to \ast into base change along the composite

    BS 1B(S 1/C p)*. B S^1 \overset{}{\longrightarrow} B (S^1/C_p) \longrightarrow \ast \,.

    But maybe I am missing something.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 24th 2017
    • (edited Jul 24th 2017)

    Is it really the case, as claimed on the page, that the S 1S^1-equivariant sphere spectrum of Blumberg-Mandell 13, example 4.9, is the same as the sphere spectrum with trivial circle action?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2017
    • (edited Jul 24th 2017)

    Notice that we are going now across the equivalence between “genuine” cyclotomic spectra, underlying which are genuine S 1S^1-equivariant spectra, and Nikolaus-Scholze cyclotomic spectra, underlying which is just a plain spectrum equipped with a plain S 1S^1-action (hence not a genuine but a “doubly naive” S 1S^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 1S^1-action. This does not mean that the genuine S 1S^1-equivariant sphere spectrum (which is a very different object altogether) is trivial in its S 1S^1-dependence.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 24th 2017

    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 1S^1-equivariant sphere spectrum has a canonical structure as a cyclotomic spectrum induced by the canonical isomorphisms…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2017

    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 genCycSpec 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 1S^1-action on the cyclotomic sphere spectrum is trivial, and I took that as wondering whether it could be true that the genuine S 1S^1-equivariant sphere spectrum somehow has trival S 1S^1-dependence. I tried to say that the first is true without implying the latter, because these S 1S^1-structures on the sphere spectrum are on the two different sides of the above equivalence, and are not “the same”.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 24th 2017

    Ah, so that’s

    Theorem II.6.9.The functor CycSp genCycSpCycSp^{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 genCycSpCycSp^{gen} \to CycSp (Prop. II.3.4) induces an equivalence between the subcategories of those objects whose underlying non-equivariant spectra are bounded below.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2017

    Okay, thanks!

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2017
    • (edited Jul 24th 2017)

    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:SpectraCycSpectra(-)^{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:XX tC pF_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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2017

    Since we’re seeing pp-completions of cyclotomic spectra, III.1.7: Δ p:X(XX) tC p\Delta_p: X \to (X \otimes \cdots \otimes X)^{t C_p}, how about other parts of the cohomology hexagon, pp-localization, etc.?

    I wonder whether the category of these spectra is the tangent (,1)(\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 AA is an S 1S^1-equivariant space together with compatible equivalences…

    Now what kind of category would these spaces form?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2017

    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 pXX^{C_p} \to X only at primes pp and sometimes at all natural numbers?

    (Need to run. Back later.)

    • CommentRowNumber14.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2017

    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

    • Christian Schlichtkrull, The cyclotomic trace for symmetric ring spectra, Geometry & Topology Monographs 16 (2009), 545–592, (pdf)

    I’m sure it can be written more nicely.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 25th 2017

    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 pC_p-s just to C p nC_{p^n}-s. Hm.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2017
    • (edited Jul 25th 2017)

    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 nn 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 Φ mn\Phi_{m n} and Φ m\Phi_m, Φ n\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?

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2017

    By the way that [BM15, Definition 4.7] they refer to, takes the term for mnm n, and then uses mn=nmm n = n m to provide two equivalences.

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 25th 2017

    p. 6

    Note that contrary to the case of orthogonal cyclotomic spectra, we do not ask for any compatibility between the maps ϕ p\phi_p for different primes pp.

    • CommentRowNumber19.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 26th 2017

    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)THH(A) of an orthogonal ring spectrum AA can be defined by evaluating the cyclic bar construction on AA or by applying B"okstedt’s original definition of THHTHH to AA. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for THH(A)THH(A). This implies that the two versions of topological cyclic homology resulting from these variants of THH(A)THH(A) are equivalent. \ ( https://arxiv.org/abs/1707.07862, 33kb)

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeApr 6th 2018

    added minimal cross-link with cycle category and cyclic set and cyclic object

    diff, v14, current

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