Chromatic Purity of Dualizable Categories

Yifan Jin

Home/Archive/arXiv

arXiv · 2609.26497

Chromatic Purity of Dualizable Categories

Yifan Jin

View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop the chromatic theory of dualizable stable categories, with chromatic purity for continuous $K$-theory at its center. We generalize the chromatic purity theorem for algebraic $K$-theory to continuous $K$-theory of dualizable stable categories and reformulate it as the purity of $H$-unital rings. Using categorical completion theory, we construct chromatic fracture squares and obtain a refinement of chromatic purity: for every dualizable stable category $\mathcal{C}$, the $T(n)\oplus T(n-1)$-completion map induces an equivalence $$ K_{T(n)}^{\mathrm{cont}}(\mathcal{C})\xrightarrow{\simeq}K_{T(n)}^{\mathrm{cont}}(\mathrm{Nuc}_{T(n)\oplus T(n-1)}(\mathcal{C})). $$ We also establish chromatic descent for continuous $K$-theory of dualizable homotopy fixed points and prove a categorical nuclear refinement of descent. This framework allows us to lift redshift bounds, Tate vanishing, and blueshift from spectra to dualizable stable categories. We show that the $T(n)$-completion of a rigid symmetric monoidal stable category is equivalent to the dualizable limit of module categories over a tower of type $n$ generalized Moore spectra. As applications, we prove that both nuclear solid and nuclear gaseous module categories satisfy chromatic redshift.

Explore related subjects

math.AT math.CT math.KT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yifan Jin. 2026-09-22. Chromatic Purity of Dualizable Categories. https://arxiv.org/abs/2609.26497

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Finite Topological Space Filtrations: A Topological Framework for Data Analysis

We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construct a sequence of coarsening topologies on the same set of points. These topologies give persistence modules and barcodes in the usual way, but they also retain information that is lost when the filtration is reduced to homology. At each level one can examine, for example, which points are topologically indistinguishable, how their minimal neighbourhoods overlap, how connected components merge, and how these features change from one level to the next. We develop the basic theory of these filtrations, establish stability results under suitable hypotheses, and give practical constructions starting directly from a distance matrix. We then study what can be learned from the resulting finite topologies. On synthetic data with known clusters of different shapes, sizes, and densities, we examine how these regions appear among the finite-topological structures and how they merge as the topology coarsens. We also study what happens when points that become uncovered early in the construction are removed and the analysis is repeated. For one-dimensional homology, we use paths in the finite-topological structure to locate cycles and to examine how their appearance is related to the geometry of the data. We finally apply these ideas to two real data sets with quite different structures. On the Paul15 single-cell data, we use the evolving finite topology to examine fine cellular states, their overlaps and relations, their assembly into larger groups, and the effect of removing points that connect these structures. On COIL20, where images of an object are sampled through a full rotation, we study how the cyclic organization of the images is reflected in the finite-topological evolution and in the associated one-dimensional homology.

math.AT↗

arXiv2026

Persistent Simple-homotopy invariants via discrete Morse theory

Persistent homology records the evolution of homological features along a filtration, but does not retain finer information related to simple-homotopy theory. In this paper, we develop two approaches to capturing such information for filtered simplicial complexes. We first introduce the Morse complexity profile, which records the minimal number of critical simplices at each filtration level. We study its invariance and stability properties and develop computable approximations using several discrete Morse matchings. We then introduce a persistent version of Whitehead torsion and show that it is invariant under both levelwise homotopy equivalence and interleaving equivalence of filtrations.

math.AT↗

arXiv2026

Signed GLMY Homology of Signed Graphs via Double Covers

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

math.AT↗