Central H-spaces and banded types
Bibliographic record
Abstract
We introduce and study central types, which are generalizations of Eilenberg–Mac Lane spaces. A type is central when it is equivalent to the component of the identity among its own self-equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product of banded types , which are the appropriate notion of torsor for a central type. Our constructions are carried out in homotopy type theory, and therefore hold in any ∞-topos. Even when interpreted into the ∞-topos of spaces, our approach to constructing these deloopings is new. Along the way, we further develop the theory of H-spaces in homotopy type theory, including their relation to evaluation fibrations and Whitehead products. These considerations let us, for example, rule out the existence of H-space structures on the 2 n -sphere for n > 0 . We also give a novel description of the moduli space of H-space structures on an H-space. Using this description, we generalize a formula of Arkowitz–Curjel and Copeland for counting the number of path components of this moduli space. As an application, we deduce that the moduli space of H-space structures on the 3-sphere is Ω 6 S 3 .
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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".