Partial evaluations and the compositional structure of the bar construction
Bibliographic record
Abstract
The algebraic expression 3 + 2 + 6 can be evaluated to 11, but it can also be partially evaluated to 5 + 6.In categorical algebra, such partial evaluations can be defined in terms of the 1-skeleton of the bar construction for algebras of a monad.We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation.The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on Set (which are those monads for which the underlying functor and the multiplication are weakly cartesian).We find that this is not true for all monads: we describe a finitary monad on Set for which the partial evaluation relation on the terminal algebra is not transitive.With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions.We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in ∆ n , but generally not all inner horns.We first of all thank Joachim Kock and an anonymous referee for detailed comments on an earlier version, which have resulted in various improvements to the exposition.This paper originates from the Applied Category Theory 2019 school.We thank the organizers
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.005 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 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".