Bibliographic record
Abstract
We show that no generalization of Whitehead's theorem holds for unpointed spaces.More precisely, we show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms.We give an explicit counterexample involving infinite symmetric groups.In contrast, we prove that the spheres do jointly reflect equivalences in the homotopy 2-category of spaces.We also show that homotopy colimits of transfinite sequential diagrams of spaces are not generally weak colimits in the homotopy category, and furthermore exhibit such a diagram with the property that none of its weak colimits is privileged, which means, roughly, that it sees the spheres as compact objects.The nonexistence of a set jointly reflecting isomorphisms in the homotopy category was originally claimed by Heller, but our results on weak colimits show that his argument had an inescapable gap, leading to the need for the new proof given here.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.004 | 0.012 |
| Open science | 0.002 | 0.010 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".