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I am thinking about the paper of Gaunce Lewis Jr. showing the incompatibility of a certain five desirable properties of spectra. This paper makes me curious about the properties of the endofunctor $Q: Grpd_0 \rightarrow Grpd_0$ of based connected spaces, defined as the connected component of the basepoint in $\text{colim } \Omega^{n}\Sigma^{n} X$. I was hoping that, if spectra are replaced with infinite loop spaces, we get more of (A1)-(A5) in the above. Specifically, I was hoping that someone could define a wedge on infinite loop spaces $\wedge^{st}$ such that:

$$ Q(X \wedge Y) \simeq^{ho} Q(X) \wedge^{st} Q(Y)$$

I am not sure about this, but it seems in keeping with the result of Freyd that infinite loop spaces are Q-algebras. In this way, I would like to consult the definition of the tensor product on an algebra for a monad (see here).

Some potentially useful observations:

  • It seems like the Freudenthal suspension theorem and the observation above (potentially enhanced to feature E${}^{\infty}$-spaces gives us that ΣQX $\simeq$ QΣX specifically.
  • It seems that, using the Freudenthal suspension theorem on $\Sigma^{n} X$, one can show that QX is weak equivalent to $\Omega \Sigma Q X$.

While I asked this question elsewhere, I am not satisfied because of the restriction to the connected component in the above, and because the paper at the link applies to spectra, not connective spectra or (based connected) infinite loop spaces.

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    $\begingroup$ The post on Mathematics: Properties of colim Ωⁿ Σⁿ X. $\endgroup$
    – Martin Sleziak
    Commented Feb 4 at 8:42
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    $\begingroup$ Your question is a point-set question, and it deserves a point set answer; since we are taking smash products and suspensions, almost certainly the equivalence does not hold for general $X$ since there are spaces for which the basepoint doesn't include as a cofibration. These point-set worries are what cause the issues, because they tend to mess up the model category behavior if you exclude them. I don't know anything about how tensor products of infinite loop spaces are defined at the point-set level, so I won't try to guess if restricting to well-based spaces solves the issue. $\endgroup$
    – Connor Malin
    Commented Feb 4 at 16:18
  • $\begingroup$ @ConnorMalin there is a result of Freyd that Q-algebras are infinite loop spaces. Perhaps there is an easier way of endowing Q-algebras with a ∧ structure? Or rather, one could observe that there is a natural map QQX ⭢ QX. $\endgroup$
    – user30211
    Commented Feb 5 at 0:30
  • $\begingroup$ It seems this question now has an answer at the MSE link above. In general, I think it would be best not to simultaneously post to both MO and MSE. Post on one, wait a week or so, then if there's no answer, post on the other platform with a link to the first thread. $\endgroup$
    – David White - gone from MO
    Commented Feb 5 at 12:50
  • $\begingroup$ @DavidWhite The questions are distinct now. The comments above were from before a change was made concerning restriction to the connected component of the "ordinary" definition of Q. $\endgroup$
    – user30211
    Commented Feb 5 at 13:31

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