Properties of colim Ωⁿ Σⁿ X

Question

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 $Q := \texttt{colim } \Omega^{n}\Sigma^{n} X$ as an endofunctor of spaces. 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 link to or confirm which of the following two are true:

• $Q$ is left adjoint to the inclusion of connective spectra into based connected spaces.

• $Q(X ∧ Y) ≃ Q(X) ∧ Q(Y)$ (homotopy equivalence)

• For an infinite loop space $X$, $ΣΩX ≃ X$ (homotopy equivalence)

The middle of these is most interesting to me. I would like to use the long exact sequence of homotopy groups for each n separately, or at the least use some kind of induction, taking colimits afterwards.

Answer 1

One note: it is a bit dubious to talk about the inclusion of infinite loop spaces into based spaces, because it is often considered more natural to remember the deloopings of your space as well. In that case, it is probably better to talk about a forgetful functor from infinite loop spaces to based spaces, which only remembers the underlying space and forgets the deloopings.

• This is true for instance $\infty$-categorically. Under the correspondence between infinite loop spaces and $\mathbb{E}_\infty$-groups (homotopy coherent commutative group objects in spaces), we want to think about $QX$ as the free commutative group on the space $X$, and therefore it makes sense to expect it to be left adjoint to the forgetful functor from $\mathbb{E}_\infty$-groups to spaces. You can formalize this in $\infty$-categories really well and it is true then (see e.g. Corollary II.32 in the lecture notes typed by Ferdinand Wagner of Fabian Hebestreit's course on algebraic and hermitian $K$-theory, and the equivalence halfway on page 100 to show that $QX\simeq\Omega^\infty\Sigma^\infty X$), but in $1$-categorical models for spectra this can be much more subtle, and I generally expect it to be false in some $1$-categorical models.
• This is false. Take for instance $X=Y=S^0$. Then $Q(X\wedge Y)=Q(S^0)\simeq \Omega^\infty\mathbb{S}$, where $\mathbb{S}$ is the sphere spectrum (as an $\Omega$-spectrum, so in the model of sequential spectra you need to fibrantly replace the usual definition of the sphere spectrum as the sequence $(S^0, S^1, S^2, \ldots)$). On the other hand, $Q(X)\wedge Q(Y)\simeq\Omega^\infty\mathbb{S}\wedge\Omega^\infty\mathbb{S}$. Now, the homotopy groups of $\Omega^\infty\mathbb{S}$ are essentially by construction the stable homotopy groups of the spheres, so $\pi_0(\Omega^\infty\mathbb{S})\cong\mathbb{Z}$. This however means that $\Omega^\infty\mathbb{S}\wedge\Omega^\infty\mathbb{S}$ has many more connected components than $\Omega^\infty\mathbb{S}$, so these spaces cannot be homotopy equivalent. The problem essentially is that while $\Sigma^\infty$ is symmetric monoidal in a homotopical sense as functor from based spaces to spectra, the functor $\Omega^\infty$ in the other direction is only lax symmetric monoidal in a homotopical sense, and $QX\simeq \Omega^\infty\Sigma^\infty X$.
• This is also false. We can for instance take $X=\mathbb{Z}\simeq K(\mathbb{Z},0)$, with its infinite loop space structure given by some choice of Eilenberg MacLane spaces $K(\mathbb{Z},n)$ for $n\geq 1$. Then $\Omega X\cong *$, so $\Sigma\Omega X\cong *$. (Note that the statement would be true if we interpret an infinite loop space as a connective spectrum and let $\Sigma$ and $\Omega$ denote the functors of the same notation in the category of all spectra. It becomes false again when $\Sigma$ and $\Omega$ denote the corresponding functors in the category of connected spectra.)

Note, as an aside, that all of the properties (A1)--(A5) in the linked paper become true (in the $\infty$-categorical sense) in the $\infty$-category of spectra. The various $1$-categories of spectra only serve as models for this object, so in a sense passing to the $\infty$-category of spectra resolves the problem that the article covers.