On the representability of actionsin a semi-abelian category
Bibliographic record
Abstract
We consider a semi-abelian category V and we write Act(G, X) for the set of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(-, X) in the case where V is locally presentable, with finite limits commuting with filtered colimits.This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra.For such categories, we prove first that the representability of Act(-, X) reduces to the preservation of binary coproducts.Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(-, X) in a general semi-abelian category.Finally, we exhibit the precise form of the more involved "if and only if" amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of "normalization of a morphism".We provide also a wide supply of algebraic examples and counter-examples, giving in particular evidence of the relevance of the object representing Act(-, X), when it turns out to exist.
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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 | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.009 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.002 | 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".