Bibliographic record
Abstract
Abstract We develop a general theory of 3-dimensional “orbifold completion”, to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category $${\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:math> as a Morita category of certain $$E_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -algebras in $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> which encode triangulation invariance. We prove that in $${\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:math> again all 1- and 2-morphisms have adjoints, that it contains $$\mathcal {T}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies $${({\mathcal {T}}_{\text {orb}})}_{\text {orb}} \cong {\mathcal {T}}_{\text {orb}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>orb</mml:mtext> </mml:msub> <mml:mo>≅</mml:mo> <mml:msub> <mml:mi>T</mml:mi> <mml:mtext>orb</mml:mtext> </mml:msub> </mml:mrow> </mml:math> . This is a categorification of the work in Carquevill and Runkel (Quantum Topol 7(2):203–279, 2016). Orbifold completion by design allows us to lift the orbifold construction from closed TQFT to the much richer world of defect TQFTs. We illustrate this by constructing a universal 3-dimensional state sum model with all defects from first principles, and we explain how recent work on defects between Witt equivalent Reshetikhin–Turaev theories naturally appears as a special case of orbifold completion.
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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 | 0.001 |
| 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 | 0.001 | 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".