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A stable 1-semiadditive category that is not 2-semiadditive

Author: Wei Yang · Added: September 26, 2026

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Abstract

For every prime $p$, we construct an object in the category of $p$-typical 1-commutative monoids in $p$-local spectra whose underlying spectrum is the $p$-complete sphere and on which the cardinality of $BC_p$ induces the zero map on underlying spectra. The construction assigns to a finite groupoid the group completion of its finite covers, with transfers that retain only components having trivial relative stabilizer. The Segal conjecture supplies the Segal condition after $p$-completion. Lee’s vanishing theorem then implies that the norm of the constant $B^2C_p$-diagram on this object is not an equivalence. This gives a counterexample to the assertion that every presentable stable 1-semiadditive category is infinitely semiadditive. We also prove that the assertion holds for compactly generated $p$-local categories, using a finite-skeleton criterion for bounded stable chromatic height, and that a compact tensor unit forces rationality.