2025/09/09 by Isabel Longbottom, Longbottom, Isabel
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Advanced Operator Algebra Research #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2509.07344
We define a class of smashing localisations which we call compactly central, and classify compactly central localisations of Sp(p) and of Sp. Our main result is that Lnf is a compactly central localisation. A map α: 1 → A in a presentably symmetric monoidal ∞-category \mathscrC is central if there exists a homotopy α⊗ idA ≃ idA ⊗ α: A → A ⊗ A. A central map α can be used to produce a smashing localisation Lα of \mathscrC, because the free 𝔼1 algebra on the 𝔼0 algebra α is an idempotent commutative algebra. When both the monoidal unit and A are compact, we call Lα compactly central. We show that when \mathscrC is (compactly generated) rigid, all compactly central localisations are finite in the sense of Miller. Not all finite localisations of Sp are compactly central. To exhibit Lnf as compactly central, we determine properties of the K(n)-homology of a map between p-local finite spectra which ensure that some tensor power of the map is central.