Homotopy Theory

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Charles Rezk

16:02

Here's something I'm failing to understand. By *classical spectrum* I mean a
sequence $X_n$ of based topological spaces together with maps
$X_n\wedge S^1\to X_{n+1}$. The thing I don't understand is how to
prove this is a model category.

All constructions I'm aware of require the following: a functor $Q$
from spectra to spectra together with a natural transformation
$\eta\colon 1\to Q$ such that (i) $\eta$ is an isomorphism on stable
homotopy groups, and (ii) $QX$ is an $\Omega$-spectrum for any $X$. I
don't understand any of the the constructions of this $(Q,\eta)$ that
I have read about.

For instance, in Mandell-May-Schwede-Shipley (MMSS), they claim to
have a construction of such a $Q$ (proof of 8.8). But their
construction uses an internal hom object in their category of spectra:
$QX$ is a *homotopy colimit* of objects $F_S(F_nS^n, X)$ in the
category. This is fine for symmetric spectra or orthogonal spectra,
but can't possibly work for classical spectra (which they call
$\mathcal{N}$-spectra), since this does not have an internal hom.

Bousfield-Friedlander (BF) mainly work simplicially, though they also
claim that there is a model structure in the topological case by
"mimicing" the simplicial construction (bottom of page 87).

BF define $QX$ (p.85) (in the simplicial version) to be the "obvious
spectrum" with $(QX)_n = \lim_{i\to\infty}
\mathrm{Sing}\Omega^i|X_{n+i}|$. They don't say what the structure
maps are, but I believe I know what they are supposed to be.

Notice BFs definition is an honest colimit, not a homotopy colimit.
What is not clear to me here is why $QX$ must be an omega spectrum:
the loop functor $\Omega$ doesn't commute with all sequential colimits
of spaces, though it does commute with sequential colimits of
inclusions. If you try to replace the honest colimit with a homotopy
colimit, I don't know how you are supposed to define the structure maps.

Probably there is some way to produce a $QX$ which works. But I don't
believe either of these as stated. Am I missing something dumb?

Harry Gindi

16:21

@CharlesRezk Dumb question, I guess, but maybe the honest colimit is a homotopy colimit because the diagram is projectively cofibrant?

Also, I think I read that the spectrification functor for sequential prespectra is not a derived left adjoint but an actual functorial fibrant replacement

so it would make sense that the colimit is not a homotopy colimit

so ignore the incorrect thing about it being a homotopy colimit

that I said in the first line

I think it's covered in this article pretty well ncatlab.org/nlab/show/…

see Proposition 0.20

The reference they give is Schwede 1997

the nLab article is really extensive. I think it gives a full proof

Tyler Lawson

17:42

@CharlesRezk I might need some caffeine but I'm not even seeing this morning that BF isn't subject to the usual "commuting two circles" problem with the structure maps

Charles Rezk

@HarryGindi I think what you point to is a solution, I'll need to look at it more

@TylerLawson It's hard to say what BF is subject to, cause they don't say what their structure map is. I'm claiming that BF don't do what you are thinking about, exactly because of the commuting two circles problem. I think BF use the isomorphism $\mathrm{colim}_{n\geq0} Y_n \approx \mathrm{colim}_{n\geq1} Y_n =\mathrm{colim}_{n\geq0} Y_{n+1}$ given by "reindexing" a sequential colimit while dropping the first term.

Thus $\mathrm{colim}_{n\geq0} \Omega^n X_n\approx \mathrm{colim}_{n\geq0} \Omega^{n+1}X_{n+1} \to \Omega \mathrm{colim}_{n\geq0} \Omega^n X_{n+1}$.

Tyler Lawson

OK. I believe that this is equivalent to what I've got written down (the maps Y_n -> Y_{n+1} definitely give a commuting ladder between the two directed sequences) and so the structure maps are the same.

Having said all of that, in BF's case I want to believe that maybe I'm supposed to commute the Ω^i outside the Sing and use that the sphere is compact in simplicial sets.

Bruno Stonek

18:56

I'm reading the discussion in Ando et al., Units of ring spectra, and I'm confused about something. The discussion in 5.3 seems to be saying that there is a left Quillen equivalence $Top \to \ast-Mod$. Then in Proposition 5.24, they say that $\Sigma^\infty_+:\ast-Mod\to \mathbb{S}-Mod$ is left Quillen. Therefore, if I compose both of these, I get a left Quillen functor $Top \to \mathbb{S}-Mod$. But * is cofibrant in Top, and $\Sigma^\infty_+(*)=\mathbb{S}$ is not cofibrant in $\mathbb{S}-Mod$.

What am I saying wrong?

2 hours later…

Harry Gindi

20:30

@BrunoStonek I have no idea the answer here, but I remember reading that "et. al." drives people here crazy

because if your name is on thr paper, it is assumed you contributed equally to the work (and almost no papers in pure math have more than four authors)

Bruno Stonek

hmm. well, that was certainly not my intention :) just to clarify, I meant the paper by Ando-Blumberg-Gepner-Hopkins-Rezk, called "units of ring spectra and Thom spectra".

Harry Gindi

nice

Bruno Stonek

I was just being lazy about typing so many names

Denis Nardin

I'm pretty sure people referred to it (when it was all in one piece instead of split up like now) as "that paper with a lot of authors about Thom spectra"

Harry Gindi

yeah but Charles Rezk was in this room earlier today and he probably could answer your question easily haha

Peter Nelson

20:35

or ABGHR

Denis Nardin

(just to be clear: I like that paper, it's just that it changed titles and had a lot of authors...)

Harry Gindi

find a mathematician whose name only contains letters in alphabetical order, collaborate with mathematicians whose names start with those letters

a good practical joke

Saul Glasman