Doubles for monoidal categories: Dedicated to Walter Tholen on his 60th birthday
Bibliographic record
Abstract
In a recent paper, Daisuke Tambara defined two-sided actions on an en- domodule (= endodistributor) of a monoidal V -category A. When A is autonomous (= rigid = compact), he showed that the V -category (that we call Tamb(A )) of so- equipped endomodules (that we call Tambara modules) is equivalent to the monoidal centre Z(A ,V ) of the convolution monoidal V -category (A ,V ). Our paper extends these ideas somewhat. For general A , we construct a promonoidal V -category DA (which we suggest should be called the double of A ) with an equivalence (DA ,V ) ' Tamb(A ). When A is closed, we define strong (respectively, left strong) Tambara modules and show that these constitute a V -category Tambs(A ) (respectively, Tambls(A )) which is equivalent to the centre (respectively, lax centre) of (A ,V ). We construct localizations DsA and DlsA of DA such that there are equivalences Tambs(A ) ' (DsA ,V ) and Tambls(A ) ' (DlsA ,V ). When A is autonomous, every Tambara module is strong; this implies an equivalence Z(A ,V ) ' (DA ,V ).
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.005 |
| Insufficient payload (model declined to judge) | 0.007 | 0.003 |
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".