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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 3rd 2016

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

(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 $sSet^{\ast/}$?

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeMar 3rd 2016

Pointed simplicial sets?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMar 4th 2016

Pointed simplicial sets. The undercategory of $sSet$ 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 $\mathcal{C}^{A/}_{/B}$ to denote the category of $\mathcal{C}$ under $A$ and over $B$. I think it’s good to have the slicing objects in small script, for $A/\mathcal{C}/B$ 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 “$A/$” come after the $\mathcal{C}$ kind of defeats the purpose of the notation $A/\mathcal{C}$.

• 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 $E$ by a super-script, i.e. “$\,E^n\,$” as opposed to the conventional subscript “$\,E_n\,$”?