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A 2-group is a 'categorified' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G G G has been replaced by a functor.Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call 'weak' and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 : xx 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 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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".