On the metrical and quantalic versions of the *-autonomous category of sup-lattices
Bibliographic record
Abstract
In 1984, Joyal and Tierney presented the category Sup of complete lattices and their suprema-preserving maps as a * -autonomous category in the sense of Barr.Work on this paper was motivated by the question whether the Joyal-Tierney proof may be extended to a metrical context, so that the order of the lattice gets replaced by a generalized metric in the sense of Lawvere.The affirmative answer we give relies crucially on working with not necessarily symmetric metrics.It applies not only to small separated and cocomplete categories enriched in the Boolean quantale 2 (reproducing Sup), or in the Lawvere quantale [0, ∞] (producing the category we were looking for), but in any commutative and unital quantale V. Benefitting from previous work by Stubbe, Hofmann, and others, with rather explicit constructions of its tensor product and the internal hom we give an alternative proof that the resulting category V-Sup is * -autonomous, a result first established by Eklund, Gutiérrez García, Höhle, and Kortelainen in 2018 from a predominantly order-theoretic perspective.
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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 | 0.003 |
| 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 | 0.007 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".