What is the homotopy category of the sphere spectrum?

Is there a known explicit description of the abelian $$2$$-group $$\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$$?

• This question looks a bit cumbersome to me. Are not you asking about the fundamental groupoid of the infinite loop space of the sphere spectrum? Sep 5, 2021 at 8:55
• @FernandoMuro Sorry, I was a bit confused.
– Théo
Sep 5, 2021 at 19:08
• The easiest description is that it is the stable quadratic module $\mathbb{Z}\otimes\mathbb{Z}\twoheadrightarrow\mathbb{Z}/(2)\stackrel{0}{\rightarrow}\mathbb{Z}$ Sep 6, 2021 at 8:31
• In case you're not acquainted with stable quadratic modules, if you want to see it as a symmetric monoidal groupoid, then the object set is $\mathbb{Z}$, the automorphism group of an object is $\{\pm1\}$, there are no morphisms other than automorphisms, the tensor product is addition on objects and multiplication on morphisms, the associativity and unitarity constraints are identities, and the commutativity constraint $m+n\rightarrow n+m$ is $(-1)^{mn}$. Sep 6, 2021 at 8:34
• @FernandoMuro I had heard of stable quadratic modules before, but at the time I had trouble finding any reference to learn about them. This time however I found a paper you wrote pointing to Baues, so I can finally understand what they are now. Thank you!
– Théo
Sep 6, 2021 at 22:55

1 Answer

This is the groupoid given by the 1-truncation $$\tau_{\leq 1}(QS^0)$$. This groupoid has $$\mathbb Z$$-many objects (since $$\pi_0^s = \mathbb Z$$), and each one has automorphism group $$C_2$$ (since $$\pi_1^s = C_2$$). The tensor product on objects is given by addition in $$\mathbb Z$$, and on morphisms by addition in $$C_2$$. One way to see this is to consider the universal functor $$\Sigma \to QS^0$$ given by the Barratt-Priddy-Quillen theorem (i.e. the fact that $$K(\Sigma) = QS^0$$; here $$\Sigma$$ is the groupoid of finite sets with the disjoint union monoidal structure), and to postcompose with the truncation functor $$QS^0 \to \tau_{\leq 1} (QS^0)$$; the fact that this functor is symmetric monoidal yields this description of the category. This perspective is discussed a bit more here.

From the description I've given, I suppose it follows that $$\tau_{\leq 1} (QS^0)$$ splits symmetric monoidally as $$\tau_{\leq 1} (QS^0) = \mathbb Z \times BC_2$$, (where $$\mathbb Z$$ is a discrete symmetric monoidal groupoid and $$BC_2$$ is a 1-object symmetric monoidal groupoid), which is maybe a little surprising. This is not to say that $$\tau_{\leq 1} \mathbb S$$ splits...

• Hi Tim, the splitting of the category is monoidal but not symmetric monoidal (that would mean you could get a model where the twist automorphism (n+m) -> (m+n) is the identity, when it should be the "(-1)^nm") Sep 5, 2021 at 1:03
• @TylerLawson Ah of course! The symmetry is the most interesting part of the story and I didn’t even address it! And it’s a good thing— contrary to what I wrote, if the equivalence were symmetric monoidal, it would give an equivalence of spectra. Sep 5, 2021 at 2:01