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nLab > Latest Changes: sequential spectrum

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 3rd 2016

    added some basics to sequential spectrum: definition, sSet */sSet^{\ast/}-enrichment, statement of equivalence to sSet */sSet^{\ast/}-enriched functors on standard spheres and of Quillen equivalence to excisive sSet */sSet^{\ast/}-functors on sSet fin */sSet^{\ast/}_{fin}.

    (This is a digest of more detailed discussion that I am typing into model structure for excisive functors.)

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 3rd 2016

    What is sSet */sSet^{\ast/}?

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 4th 2016

    Pointed simplicial sets?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2016

    Pointed simplicial sets. The undercategory of sSetsSet under the point.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 4th 2016

    Is the reason for placing */\ast/ in a superscript mainly aesthetic?

    I’m not sure how self-explanatory the notation is. (I thought it might be pointed simplicial sets, but wasn’t sure.) So I’ll insert a parenthetical unless it’s already been done. I think there are other cases where you (Urs) put things in superscripts whose meaning was not immediately clear to me, so I may get back to you on that.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2016

    I’ll insert a parenthetical

    Thanks. I had been stating my notation conventions in various related entries that I had been editing, but apperently not in this one. Thanks for catching this.

    Regarding choice of notation: I use 𝒞 /B A/\mathcal{C}^{A/}_{/B} to denote the category of 𝒞\mathcal{C} under AA and over BB. I think it’s good to have the slicing objects in small script, for A/𝒞/BA/\mathcal{C}/B looks clunky and becomes misleading at least once the objects have longer names than the category.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMar 4th 2016

    On the other hand, having the “A/A/” come after the 𝒞\mathcal{C} kind of defeats the purpose of the notation A/𝒞A/\mathcal{C}.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2016
    • (edited Mar 4th 2016)

    Hm, are we really to have a fight about trivial choices in notation here? I’d rather not. I have been using this notation for ages, other people have, too, it’s no worse than any other choice.

    What, though, would be a good source for Quillen equivalence between the Bousfield-Friedlander stable model structure and the stable model structure on symmetric spectra?

    • CommentRowNumber9.
    • CommentAuthorKarol Szumiło
    • CommentTimeMar 4th 2016

    There is a comparison in Chapter 4 of Symmetric Spectra by Hovey, Shipley, Smith.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2016

    Thanks!

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2016
    • (edited Apr 22nd 2016)

    There is a slightly subtle interplay between three different models for the looping and suspension operation on sequential spectra (with analogues on highly structured spectra) – “real suspension” and “fake suspension” (alas) and shifting. It seems that one of the few places whith a comprehensive account of this is J. F. Jardine’s recently published book Local homotopy theory. I have started a section of this in the entry (here).

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2021

    Trivia question:

    Is there any author who indexes component spaces of a spectrum EE by a super-script, i.e. “E n\,E^n\,” as opposed to the conventional subscript “E n\,E_n\,”?

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