I am trying to understand the following statement, which is taken from Appendix B, Proposition B.12 of Hopkins and Singer's paper:
The correspondence $$ E\mapsto \pi_{\leq 1}E_0 $$ is an equivalence between the 2-category of spectra E satisfying $\pi_iE=0, i\neq 0, 1$ and the 2-category of Picard categories.
In a previous definition (Definition B.5), they defined the k-invariant of a Picard category $C$ as an abelian group homomorphism $$\pi_0C\otimes\mathbb{Z}/2\rightarrow \pi_1C,$$ which is induced by a "twist" on the tensor product. Also, they had made $\pi_{\leq 1}E_n$ into a Picard category in Example B.7 (see my previous question).
In the proof of the above proposition, it seems to me they were relating the following two concepts of k-invariants:
- k-invariant for Postnikov tower of spectra
- k-invariant for Picard category
More precisely, for a spectrum $E$ with only $\pi_0,\pi_1$ nontrivial, its k-invariant (from the Postnikov tower of $E$) $$ H\pi_0E\rightarrow \Sigma^2H\pi_1E$$ determines its homotopy type. Then they claimed that
The set of homotopy classes of maps $ H\pi_0E\rightarrow \Sigma^2H\pi_1E$ is naturally isomorphic to $$\mathrm{hom}(A\otimes\mathbb{Z}/2, B).$$
without pointing out what $A$ and $B$ are referring to. After that, they concluded the proof by relating the two concepts of k-invariant:
One can associate to a Picard category with k-invariant $\epsilon$, the spectrum E with the same k-invariant.
I am quite confused by their proof. Especially the part involving $A$ and $B$, also how they transfer the k-invariant of Postnikov tower to the k-invariant of Picard category.
Thank you in advance for any comments, answers, and corrections!
$A$ and $B$ are just $\pi_0$ and $\pi_1$; they’re citing a nontrivial but old theorem of stable homotopy theory in moving to maps out of $A\otimes \mathbb Z/2.$
I’d suggest reading an actual proof of this result, which I believe wasn’t done until Johnson and Osorno here: https://arxiv.org/pdf/1201.2686.pdf