Bibliographic record
Abstract
The purpose of this writing is to explore the exact relationship running between geometric infinity-toposes and Mike Shulman's proposal for the notion of elementary 1-topos, and in particular we will focus on the set-theoretical strength of Shulman's axioms, especially on the last one dealing with dependent sums and products, in the context of geometric infinity-toposes. Heuristically, we can think of a collection of morphisms which has a classifier and is closed under these operations as a well-behaved internal universe in the infinity-category under consideration. We will show that this intuition can in fact be made to a mathematically precise statement, by proving that, once fixed a Grothendieck universe, the existence of such internal universes in geometric infinity-toposes is equivalent to the existence of smaller Grothendieck universes inside the bigger one. Moreover, a perfectly analogous result can be shown if instead of geometric infinity-toposes our analysis relies on ordinary sheaf toposes, although with a slight change due to the impossibility of having true classifiers in the infinity-dimensional setting. In conclusion, it will be shown that, under stronger assumptions positing the existence of intermediate-size Grothendieck universes, examples of elementary infinity-toposes with strong universes which are not geometric can be found.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".