Bibliographic record
Abstract
We give necessary and sufficient conditions on a presentable ∞-category C so that families of objects of C form an ∞-topos.In particular, we prove a conjecture of Joyal that this is the case whenever C is stable.Let X be an ∞-topos and let C be a sheaf of ∞-categories on X.We denote by X C → X the cartesian fibration classified by C.An object in X C is thus a pair (U, c) with U ∈ X and c ∈ C(U ).The question we are interested in is the following:If C is a presentable ∞-category, we will also denote by X C → X the cartesian fibration classified by the sheafJoyal calls C an ∞-locus if S C is an ∞-topos, 1 and he conjectures that any presentable stable ∞-category is an ∞-locus [2].The motivating example, due to Biedermann and Rezk, is the ∞-category Sp of spectra: there is an equivalenceand the right-hand side is an ∞-topos [3, Remark 6.1.1.11].More generally, for any small ∞-category A with finite colimits and a final object,is an ∞-topos.This paper gives a partial answer to Question 1.1 in Theorem 1.2.As a corollary, we obtain a characterization of ∞-loci (see Corollary 1.5), similar to Rezk's characterization of ∞-topoi, which easily implies Joyal's conjecture (see Example 1.7).The latter also follows more directly from two observations, applicable to any Pr R -valued sheaf C on an ∞-topos X:
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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 | 0.003 |
| 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 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 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".