Model structures for homotopy ofinternal categories
Bibliographic record
Abstract
The aim of this paper is to describe Quillen model category structures on the category CatC of internal categories and functors in a given finitely complete category C. Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on C.Under mild conditions on C, the regular epimorphism topology determines a model structure where we is the class of weak equivalences of internal categories (in the sense of Bunge and Paré).For a Grothendieck topos C we get a structure that, though different from Joyal and Tierney's, has an equivalent homotopy category.In case C is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and the model structure on CatC induces a notion of homotopy of internal crossed modules.In case C is the category Gp of groups and homomorphisms, it reduces to the case of crossed modules of groups.The trivial topology on a category C determines a model structure on CatC where we is the class of strong equivalences (homotopy equivalences), fib the class of internal functors with the homotopy lifting property, and cof the class of functors with the homotopy extension property.As a special case, the "folk" Quillen model category structure on the category Cat = CatSet of small categories is recovered.Contents 1 Introduction 67 2 Preliminaries 69 3 A cocylinder on CatC 74 4 T -equivalences 77 5 The T -model structure on CatC 80 6 Case study: the regular epimorphism topology 85 7 Case study: the trivial topology 90 The first author's research is financed by a Ph.D. grant of the Institute of Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.008 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 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".