Bibliographic record
Abstract
Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits.We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions.We also expand on an undeservedly ignored theorem of Kennison and Gildenhuys: that the codensity monad of the inclusion of (finite sets) into (sets) is the ultrafilter monad.This result is analogous to the correspondence between measures and integrals.So, for example, we can speak of integration against an ultrafilter.Using this language, we show that the codensity monad of the inclusion of (finite-dimensional vector spaces) into (vector spaces) is double dualization.From this it follows that compact Hausdorff spaces have a linear analogue: linearly compact vector spaces.Finally, we show that ultraproducts are categorically inevitable: the codensity monad of the inclusion of (finite families of sets) into (families of sets) is the ultraproduct monad.Now we have at last obtained permission to ventilate the facts. . .
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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 | 0.002 |
| 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".