Bibliographic record
Abstract
This article is the rst part of a series of three articles, in which we develop a higher covering theory of racks and quandles.This project is rooted in M. Eisermann's work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are central, relatively to trivial quandles.We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions dening our higher coverings of racks and quandles.In this rst article (Part I), we revisit the foundations of the covering theory of interest, we extend it to the more general context of racks and mathematically describe how to navigate between racks and quandles.We explain the algebraic ingredients at play, and reinforce the homotopical and topological interpretations of these ingredients.In particular we study and insist on the crucial role of the left adjoint of the conjugation functor Conj between groups and racks (or quandles).We rename this functor Pth, and explain in which sense it sends a rack to its group of homotopy classes of paths.We characterize coverings and relative centrality using Pth, but also develop a more visual geometrical understanding of these conditions.We use alternative generalizable and visual proofs for the characterization of central extensions of racks and quandles.We complete the recovery of M. Eisermann's suitable constructions of weakly universal covers, and fundamental groupoids from a Galois-theoretic perspective.We sketch how to deduce M. Eisermann's detailed classication results from the fundamental theorem of categorical Galois theory.As we develop this complementary understanding of the subject, we lay down all the ideas and results which will articulate the higher-dimensional theory developed in Part II and III.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 0.000 |
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".