Bibliographic record
Abstract
A 2-group is a 'categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G × G → G has been replaced by a functor.Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call 'weak' and 'coherent' 2-groups.A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a 'weak inverse': an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x.A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x and isomorphisms i x : 1 → x⊗ x, e x : x⊗x → 1 forming an adjunction.We describe 2-categories of weak and coherent 2-groups and an 'improvement' 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories.We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups.We give a tour of examples, including the 'fundamental 2-group' of a space and various Lie 2-groups.We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology.Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G ( ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object.These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g ( ∈ R) described in a companion paper.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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".