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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, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , that map a Hopf monoid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> to graded Hopf algebras <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒦</mml:mi> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , respectively. There is a natural Hopf monoid structure on linear orders <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">L</mml:mi> </mml:math> , and the two Fock functors are related by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒦</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Unlike the functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , the functor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> applied to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> may not preserve the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> . In view of the relation between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒦</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , one may consider instead of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> the larger Hopf monoid <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> and study the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> . One of the main results in this paper provides a cancellation free and multiplicity free formula for the antipode of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">L</mml:mi> <mml:mo>×</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:math> . As a consequence, we obtain a new antipode formula for the Hopf algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mi>𝒦</mml:mi> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We explore the case when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">H</mml:mi> </mml:math> is commutative and cocommutative, and obtain new antipode formulas that, although not cancellation free, they can be used to obtain an antipode formula for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>𝒦</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="bold">H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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 ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> )-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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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 it