The Shuffle Quasimonad and Modules with Differentiation and Integration
Bibliographic record
Abstract
Differential linear logic and the corresponding categorical structure, differential categories , introduced the idea of differential structure associated to a (co)monad. Typically in settings such as algebraic geometry, one expresses differential structure for an algebra by having a module with a derivation, i.e. a map satisfying the Leibniz rule. In the monadic approach, we are able to continue to work with algebras and derivations, but the additional structure allows us to define other rules of the differential calculus for such modules; in particular one can define a monadic version of the chain rule as well as other basic identities. In attempting to develop a similar theory of integral linear logic, we were led to consider the shuffle multiplication . This was shown by Guo and Keigher to be fundamental in the construction of the free Rota-Baxter algebra , the Rota-Baxter equation being the integral analogue of the Leibniz rule. This shuffle multiplication induces a quasimonad on the category of vector spaces. The notion of quasimonad, called r -unital monad by Wisbauer, is slightly weaker than that of monad, but is still sufficient to define a sensible notion of module with differentiation and integration. In this paper, we demonstrate this quasimonad structure, show that its free modules have both differential and integral operators satisfying the Leibniz and Rota-Baxter rules and satisfy the fundamental theorems of calculus.
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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 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".