The more legs the merrier: A new composition for symmetric (multi-)lenses
Bibliographic record
Abstract
Lenses are a category theoretic construct and are used in a wide variety of applications. Symmetric lenses compose to, of course, form new symmetric lenses. Symmetric lenses are usually represented as spans of asymmetric lenses. In many applications, the fact that a symmetric lens might also be represented as a cospan of asymmetric lenses is important, especially for implementation purposes. However, the composition of symmetric lenses does not preserve the property that the lenses can be represented by cospans -- two such symmetric lenses may (and frequently do) compose to form a symmetric lens which cannot be represented as a cospan of asymmetric lenses. Thus preserving the factorisation to show how cospans of asymmetric lenses might be used in implementations becomes important. In 2018, the first work on multilenses was begun. Multilenses can be represented as multi-spans of asymmetric lenses (often called 'wide spans', these are spans with an arbitrary finite number of legs). In this paper we analyse a small but realistic example of a supply chain in which the cospan representations would be 'composed away' by ordinary symmetric lens composition, and introduce a new kind of composition which we call 'fusion' in which two ordinary symmetric lenses (spans with two legs) fuse to form a multilens with three legs preserving the cospan representations, and more generally, two symmetric multilenses, spans with say m and n legs, fuse to form a symmetric multilens with m+n-1 legs, again preserving cospan representations.
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 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".