Symmetric-monoidal-associative smash product up to homotopy

Question

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.

Recall that a sequential spectrum consists of a sequence of based connected spaces $X_n$ with maps $X_n \rightarrow \Omega Y_n$, or equivalently maps $\Sigma X_n \rightarrow Y_n$.

In searching for a convenient construction of the stable homotopy category which features more of the mutually exclusive properties established here, I thought of the concept of a "pre-map":

Premaps: Here I would like to use a "pre-map" of spectra. To define this, let X and Y be sequential spectra. A pre-map $f : X \rightarrow Y$ consists of maps $f : X_n \rightarrow Y_n$ which produce commutative diagrams involving the structure maps in the stable homotopy category.

The rationale is that, since the stable homotopy category has all of the five properties in the paper linked above, these can be lifted, not to a map, but at least to a premap. This more lax condition could remove the foundational troubles associated to wedge with $S^1$ and the related coherence.

The main things I wanted to try first is showing that sequential spectra with premaps forms a symmetric monoidal category:

Symmetry: To define pre-symmetry, let $X$ and $Y$ be sequential spectra. I would like to construct a premaps $f : X \wedge Y \rightarrow Y \wedge X$ and $g : Y \wedge X \rightarrow X \wedge Y$.

Preassociativity: To define pre-symmetry, let $X$ and $Y$ be sequential spectra. I would like to construct a premaps $f : X \wedge (Y \wedge Z) \rightarrow (X \wedge Y) \wedge Z$ and $g : (X \wedge Y) \wedge Z \rightarrow X \wedge (Y \wedge Z)$.

In sum, I imagine a category of spectra whose hom component consists of these premaps instead, and I suspect that one can always lift the desirable properties of the stable homotopy category to the category of sequential spectra with premaps as maps. I'd love to hear about any convenience that ensures, particularly the five properties (A1)-(A5) in the paper by Gaunce Lewis Jr. linked to above.

Answer 1

We don't just have a problem with commutativity of the structure maps in the stable category; there are also not enough premaps to construct a map $\tau$ representing the twist automorphism in the stable category. $\newcommand{\SS}{\mathbb{S}}$

For any $n$, there is a sequential spectrum $\SS^{-n}$ such that $$ (\SS^{-n})_k = \begin{cases} \ast &\text{if }k < n,\\ S^{k-n} &\text{if }k \geq n, \end{cases} $$ with structure maps being isomorphisms in degrees $n$ and above. These satisfy $\SS^{-n} \wedge \SS^{-m} \simeq \SS^{-(n+m)}$ (isomorphic using actual maps of sequential spectra, and hence also using premaps). These represent the negative spheres in the stable homotopy category.

Any premap $\SS^{-n} \to \SS^{-n}$ consists of maps $f_k: S^{k-n} \to S^{k-n}$. For $k \geq n$, the premap condition asks that $\Sigma f_k$ and $f_{k+1}$ be equivalent in the stable homotopy category. But maps between spheres are completely classified, up to (stable) homotopy, by their degree, and so being a premap is the same as asking all of the $f_k$ to have the same degree.

However, the map $f_n$ is a (pointed) map $S^0 \to S^0$, and this can only have degree 0 or 1. So premaps $\SS^{-n} \to \SS^{-n}$ can only have degree 0 or 1.

If we choose an isomorphism $\SS^{-n} \wedge \SS^{-n} \to \SS^{-(2n)}$, any choice of $\tau: \SS^{-n} \wedge \SS^{-n} \to \SS^{-n} \wedge \SS^{-n}$ becomes isomorphic to a self-map of $\SS^{-(2n)}$, and this means that the only choice is for it to have degree 1. The twist self-map of $\SS^{-n}$ in the stable homotopy category, however, has degree $(-1)^n$.

(If you ask for $\tau$ to be natural in $X$ and $Y$, you can prove that $\tau$ can't exist.)

One intuition from this is that sequential spectra are based on $\mathbb{N}$, which is "too symmetric" for the stable homotopy category.

Different approaches to solving Lewis' objections have solved this in different ways.

• Adams used a "cells now, maps later" approach to ensure that enough maps were defined.
• $\SS$-modules (EKMM) imposed a "spectrification" / $\Omega$-spectrum condition so that any map in the stable homotopy category could automatically be realized by a map/premap.
• Symmetric spectra (HSS) deliberately added extra symmetry isomorphisms.
• Spectra as in Higher Algebra (Lurie) also imposes an $\Omega$-spectrum condition.
• It's also possible to, instead, use even spheres: use S^2 instead of S^1.