Bibliographic record
Abstract
In this paper, a Hopf monoid is an algebraic structure built on objects in the category of Joyal’s vector species. There are two Fock functors, 𝒦 and 𝒦 ¯ , that map a Hopf monoid H to graded Hopf algebras 𝒦 ( H ) and 𝒦 ¯ ( H ) , respectively. There is a natural Hopf monoid structure on linear orders L , and the two Fock functors are related by 𝒦 ( H ) = 𝒦 ¯ ( H × L ) . Unlike the functor 𝒦 ¯ , the functor 𝒦 applied to H may not preserve the antipode of H . In view of the relation between 𝒦 and 𝒦 ¯ , one may consider instead of H the larger Hopf monoid L × H and study the antipode of L × H . One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of L × H . As a consequence, we obtain a new antipode formula for the Hopf algebra H = 𝒦 ( H ) . We explore the case when H is commutative and cocommutative, and obtain new antipode formulas that, although not cancellation free, they can be used to obtain an antipode formula for 𝒦 ¯ ( H ) in some cases. We also recover many well-known identities in the literature involving antipodes of certain Hopf algebras. In our study of commutative and cocommutative Hopf monoids, hypergraphs and acyclic orientations play a central role. We obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanley’s ( - 1 )-color theorem. An important consequence of our notion of acyclic orientation of hypergraphs is a geometric interpretation for the antipode formula for hypergraphs. This interpretation, which differs from the recent work of Aguiar and Ardila as the Hopf structures involved are different, appears in subsequent work by the authors.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.000 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".