Bibliographic record
Abstract
Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street's weak wreath product construction.That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl (n) (K), of (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation.The first instance Wdl (0) (K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl (n) (K) contains Mnd n+1 (K) as a full 2-subcategory.For the local idempotent closure K of K, extending the multiplication of the 2-monad Mnd, we equip these 2-categories with n possible 'weak wreath product' 2-functors Wdl (n) (K) Wdl (n-1) (K), such that all of their possible n-fold composites Wdl (n) (K) Wdl (0) (K) are equal; that is, such that the weak wreath product is 'associative'.Whenever idempotent 2cells in K split, this leads to pseudofunctors Wdl (n) (K) Wdl (n-1) (K) obeying the associativity property up-to isomorphism.We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras.We also construct a fully faithful embedding of Wdl (n) (K) into the 2-category of commutative n + 1 dimensional cubes in Mnd(K) (hence into the 2-category of commutative n + 1 dimensional cubes in K whenever K has Eilenberg-Moore objects and its idempotent 2-cells split).Finally we give a sufficient and necessary condition on a monad in K to be isomorphic to an n-ary weak wreath product.
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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 | 0.000 |
| 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".