The category of extensions and idempotent completion
Bibliographic record
Abstract
Building on previous work, we study the splitting of idempotents in the category of extensions E -Ext(C) associated to a pair (C, E) of an additive category and a biadditive functor to the category of abelian groups.In particular, we show that idempotents split in E -Ext(C) whenever they do so in C, allowing us to prove that idempotent completions and extension categories are compatible constructions in a 2-category-theoretic sense.Furthermore, we show that the exact category obtained by first taking the idempotent completion of an n-exangulated category (C, E, s), in the sense of Klapproth-Msapato-Shah, and then considering its category of extensions is equivalent to the exact category obtained by first passing to the extension category and then taking the idempotent completion.These two different approaches yield a pair of 2-functors each taking small n-exangulated categories to small idempotent complete exact categories.The collection of equivalences that we provide constitutes a 2-natural transformation between these 2-functors.Similar results with no smallness assumptions and regarding weak idempotent completions are also proved.The authors are grateful to Dixy Msapato for directing them to a result in their paper [Msapato, 2022], which led to Proposition 4.2.Parts of this work were carried out while the second author visited Aarhus University, and while the first and fourth author visited
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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 | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.001 |
| 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".