The Sphere of Semiadditive Height 1

Abstract

Abstract We construct a lift of the $p$-complete sphere to the universal height $1$ higher semiadditive stable $\infty $-category of Carmeli–Schlank–Yanovski, providing a counterexample, at height $1$, to their conjecture that the natural functor $ _n \to \operatorname{\textrm{Sp}}_{T(n)}$ is an equivalence. We then record some consequences of the construction, including an observation of Schlank that this gives a conceptual proof of a classical theorem of Lee on the stable cohomotopy of Eilenberg–MacLane spaces.