Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I gave spectrification its own entry, in order to collect in one place various constructions such as 1-excisive reflection, Joyal’s parameterized sequential spectrification, as well as Lewis-May-Steinberger’s original “polemical definition”.
In the diagram in Definition 1, shouldn’t n∈Z actually be n≥0? (Spectra are indexed by natural numbers, not by integers, or so I thought.)
Can this be made to work in the case of symmetric spectra? I can imagine adding Σ_n-arrows to each diagonal vertex in the diagram (with the new compositions being uniquely defined).
Would this give rise to a spectrification functor for symmetric spectra?
In the diagram in Definition 1, shouldn’t n∈Z actually be n≥0? (Spectra are indexed by natural numbers, not by integers, or so I thought.)
It gives equivalent categories either way, but of course you are right that indexing over is standard. I have changed it in the entry.
Can this be made to work in the case of symmetric spectra?
Here I am not sure what you have in mind when you say “this”, as clearly the exact component construction does not apply, while just as of course there is spectrification also for symmetric spectra.
while just as of course there is spectrification also for symmetric spectra.
I mean the general statement in Lemma 1, not the specific construction in the section “For sequential spectra”.
I remember reading (in Schwede’s book?) that an analog of the formula in the section “For sequential spectra” for symmetric spectra is not known, so I was wondering if Joyal’s general formalism can be exploited to produce such an explicit formula for the spectrification of symmetric spectra.
That formula probably does not work, but some stand-in does. For instance there is a good model structure on symmetric spectra for which the fibrant objects are the (injective) symmetric -spectra. So any fibrant replacement functor here will serve as spectrification.
@Urs: Yes, of course, but the question is whether this fibrant replacement functor can be described reasonably explicitly (as opposed to a small object argument construction), just like the fibrant replacement functor for nonsymmetric spectra can be described explicitly.
Ah, I see. So I don’t know, but if I come across anything, I’ll drop you a note.
back to #6, #7:
spectrification for symmetric spectra is discussed in Schwede’s Symmetric spectra around prop. 4.39, corollary 4.40.
Thanks, this is quite interesting!
1 to 9 of 9