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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.)
What is $sSet^{\ast/}$?
Pointed simplicial sets?
Pointed simplicial sets. The undercategory of $sSet$ under the point.
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.
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.
On the other hand, having the “$A/$” come after the $\mathcal{C}$ kind of defeats the purpose of the notation $A/\mathcal{C}$.
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?
There is a comparison in Chapter 4 of Symmetric Spectra by Hovey, Shipley, Smith.
Thanks!
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).
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 “$\,E_n\,$”?
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