Higher Dimensional Algebra VII: Groupoidification
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Bibliographic record
Abstract
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids.We introduce this idea with a detailed exposition of 'degroupoidification': a systematic process that turns groupoids and spans into vector spaces and linear operators.Then we present three applications of groupoidification.The first is to Feynman diagrams.The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections.This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams.The second application is to Hecke algebras.We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power.We illustrate this with the simplest nontrivial example, coming from the A 2 Dynkin diagram.In this example we show that the solution of the Yang-Baxter equation built into the A 2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field F q .The third application is to Hall algebras.We explain how the standard construction of the Hall algebra from the category of F q representations of a simply-laced quiver can be seen as an example of degroupoidification.This in turn provides a new way to categorify-or more precisely, groupoidify-the positive part of the quantum group associated to the quiver.
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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 it