Bibliographic record
Abstract
As George Bergman pointed out at the International Conference of Mathematicians in Vancouver (1974) category theory can be a very efficient way to determine all possible operations on sets. In fact, under certain conditions the operations on the values of a (representable) set valued functor are in one to one correspondence with the cooperations on the representing object and the relations that hold between those operations can be determined by analyzing the relations that hold between the cooperations on the representing object. In this thesis the ideas of Bergman described above together with the approaches described below are applied to the group of units of an arbitrary ring and the idempotent elements of commutative rings. Algebra, one of the oldest and still most popular branches of mathematics, is the study of algebraic structures. An algebraic structure, loosely speaking, is a object together with one or more operations on it. If there is more than one operation on the set these operations often satisfy certain relations. The class of objects which share certain properties is called a category. Some of the most common such categories consist of objects which posses an underlying set (e.g. groups, rings, fields, vector spaces). Often these sets can be regarded as members of different categories and accordingly are endowed with a number of operations between which relations hold. In the case that a particular object is part of different categories, one can try (more often than not successfully) to make use of certain facts that are true in one category to gain knowledge about other objects in the second category. One other approach to analyze algebraic structures is to look at their substructures (whether those lie within this category or in a different one) and gain knowledge about the original object by composing the pieces of the puzzle gained by analyzing the sub objects. Frequently the opposite approach works quite as well, i.e. to look at the objects which contain an object as subobjects.
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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 | 0.005 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.005 | 0.014 |
| Open science | 0.001 | 0.008 |
| Research integrity | 0.001 | 0.003 |
| 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".