Covering Dimension of C*-Algebras and 2-Coloured Classification
Bibliographic record
Abstract
We introduce the concept of finitely coloured equivalence for unital ∗ ^* -homomorphisms between C ∗ \mathrm C^* -algebras, for which unitary equivalence is the 1 1 -coloured case. We use this notion to classify ∗ ^* -homomorphisms from separable, unital, nuclear C ∗ \mathrm C^* -algebras into ultrapowers of simple, unital, nuclear, Z \mathcal {Z} -stable C ∗ \mathrm C^* -algebras with compact extremal trace space up to 2 2 -coloured equivalence by their behaviour on traces; this is based on a 1 1 -coloured classification theorem for certain order zero maps, also in terms of tracial data. As an application we calculate the nuclear dimension of non-AF, simple, separable, unital, nuclear, Z \mathcal {Z} -stable C ∗ \mathrm C^* -algebras with compact extremal trace space: it is 1. In the case that the extremal trace space also has finite topological covering dimension, this confirms the remaining open implication of the Toms-Winter conjecture. Inspired by homotopy-rigidity theorems in geometry and topology, we derive a “homotopy equivalence implies isomorphism” result for large classes of C ∗ \mathrm {C}^{*} -algebras with finite nuclear dimension.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".