Bibliographic record
Abstract
In the category of sets and partial functions, PAR, while the disjoint union is the usual categorical coproduct, the Cartesian product becomes a restriction categorical analogue of the categorical product: a restriction product.Nevertheless, PAR does have a usual categorical product as well in the form A&B = A B (A B).Surprisingly, asking that a distributive restriction category (a restriction category with restriction products and coproducts ) has A&B a categorical product is enough to imply that the category is a classical restriction category.This is a restriction category which has joins and relative complements and, thus, supports classical Boolean reasoning.The first and main observation of the paper is that a distributive restriction category is classical if and only if A&B = A B (A B) is a categorical product in which case we call & the "classical" product.In fact, a distributive restriction category has a categorical product if and only if it is a classified restriction category.This is in the sense that every map A / / B factors uniquely through a total map A / / B 1, where 1 is the restriction terminal object.This implies the second significant observation of the paper, namely, that a distributive restriction category has a classical product if and only if it is the Kleisli category of the exception monad 1 for an ordinary distributive category.Thus having a classical product has a significant structural effect on a distributive restriction category.In particular, the classical product not only provides an alternative axiomatization for being classical but also for being the Kleisli category of the exception monad on an ordinary distributive category.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".