Bibliographic record
Abstract
This paper studies a category X with an endofunctor T : X / / X .A T -algebra is given by a morphism T X / / X in X .We examine the related questions of when T freely generates a triple (or monad) on X ; when an object X X freely generates a T -algebra; and when the category of T -algebras has coequalizers and other colimits.The paper defines a category of "T -horns" which effectively contains X as well as all T -algebras.It is assumed that X is cocomplete and has a factorization system (E , M ) satisfying reasonable properties.An ordinal-indexed sequence of T -horns is then defined which provides successive approximations to a free T -algebra generated by an object X X , as well as approximations to coequalizers and other colimits for the category of T -algebras.Using the notions of an M -cone and a separated T -horn it is shown that if X is M -well-powered, then the ordinal sequence stabilizes at the desired free algebra or coequalizer or other colimit whenever they exist.This paper is a successor to a paper written by the first author in 1970 that showed that T generates a free triple when every X X generates a free T -algebra.We also consider colimits in triple algebras and give some examples of functors T for which no X X generates a free T -algebra.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".