Categorical Models of the Differential λ-Calculus Revisited
Bibliographic record
Abstract
The paper shows that the Scott-Koymans theorem for the untyped λ -calculus extends to the differential λ -calculus. The main result is that every model of the untyped differential λ -calculus may be viewed as a differential reflexive object in a Cartesian closed differential category. This extension of the Scott-Koymans theorem depends critically on unravelling the somewhat subtle issue of which idempotents can be split so that differential structure lifts to the idempotent splitting. The paper uses (total) Turing categories with “canonical codes” as the basic categorical semantics for the λ -calculus. It shows how the main result may be developed in a modular fashion by first adding left-additive structure to a Turing category, and then – on top of that – differential structure. For both levels of structure it is necessary to identify how “canonical codes” behave with respect to the added structure and, furthermore, how “universal objects” behave. The latter is closely tied to the question – which is the crux of the paper – of which idempotents can be split in these more structured settings.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.004 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".