Bibliographic record
Abstract
Let K be a locally finitely presentable category. If K is abelian and the sequence 0 K // X // k // C c // // 0 // is short exact, we show that 1) K is finitely generated⇔ c is finitely presentable; 2) k is finitely presentable⇔ C is finitely presentable. The “⇐” directions fail for semi-abelian varieties. We show that all but (possibly) 2)(⇐) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(⇐), it holds as soon as K is also co-homological, and all its strong epimorphisms are regular. Finally, locally finitely coherent (resp. noetherian) abelian categories are characterized as those for which all finitely presentable morphisms have finitely generated (resp. presentable) kernel objects. 1. Finitely presentable morphisms Recall (from [GU, 71] or [AR, 94]) that an object X in a category K is finitely presentable (finitely generated) if the hom-functor K(X,−) : X −→ Set preserves filtered colimits (resp. colimits of filtered diagrams made of monomorphisms). Then, K is finitely accessible if it has a (small) set of finitely presentable objects whose closure under filtered colimits is all of K. Finally, K is locally finitely presentable if it is finitely accessible and cocomplete. 1.1. Definition. Let f : X // Y be a morphism in K. (a) f is finitely presentable (resp. finitely generated) if it is a finitely presentable (resp. finitely generated) object of the slice category (X ↓ K). (b) f is finitary if X and Y are finitely presentable. The finitary morphisms of K are actually the finitely presentable objects of the category of morphisms K→. They are precisely those finitely presentable morphisms with a finitely presentable domain (see below). Finitely presentable morphisms have been first considered in algebraic geometry, where they play an important role (for example in the Chevalley Theorem; see [GD, 64] and [D, 92]). In fact, in the category CRng of commutative rings, the definition above Received by the editors 2010-11-13 and, in revised form, 2010-05-03. Transmitted by J. Rosicky. Published on 2010-05-06. 2000 Mathematics Subject Classification: 18A20, 18E10, 18C35, 18E15.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.003 |
| 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".