Strongly Hopfian and Strongly Cohopfian Objects in the Category of Complexes of Left $A$-Modules
Bibliographic record
Abstract
The object of this paper is the study of \emph{strongly hopfian}, \emph{strongly cohopfian}, \emph{quasi-injective}, \emph{quasi-projective}, \emph{Fitting} objects of the category of complexes of $A$-modules.In this paper we demonstrate the following results:a)If $C$ is a strongly hopfian chain complex (respectively strongly cohopfian chain complex) and $E$ a subcomplex which is direct summand then $E$ and $C/E$ are both strongly Hopfian (respectively strongly coHopfian) then $C$ is strongly Hopfian (respectively strongly coHopfian).b)Given a chain complex $C$, if $C$ is quasi-injective and strongly-hopfian then $C$ is strongly cohopfian.c)Given a chain complex $C$, if $C$ is quasi-projective and strongly-cohopfian then $C$ is strongly hopfian.In conclusion the main result of this article is the following theorem:Any \emph{quasi-projective} and \emph{strongly-hopfian} or \emph{quasi-injective} and \emph{strongly-cohofian} chain complex of $A$-modules is a \emph{Fitting} chain complex.
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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 | 0.006 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".