Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters
Bibliographic record
Abstract
This paper clarifies basic definitions in the universal construction of topological theories and monoidal categories.The definition of the universal construction is given for various types of monoidal categories, including rigid and symmetric.It is also explained how to set up the universal construction for non-monoidal categories.The second part of the paper explains how to associate a rigid symmetric monoidal category to a small category, a sort of the Brauer envelope of a category.The universal construction for the Brauer envelopes generalizes some earlier work of the first two authors on automata, power series and topological theories.Finally, the theory of pseudocharacters (or pseudo-representations), which is an essential tool in modern number theory, is interpreted via one-dimensional topological theories and TQFTs with defects.The notion of a pseudocharacter is studied for the Brauer categories and the lifting property to characters of semisimple representations is established in characteristic 0 for the Brauer categories with at most countably many objects.The paper contains a brief discussion of pseudo-holonomies, which are functions from loops in a manifold to real numbers similar to traces of the holonomies along loops of a connection on a vector bundle on the manifold.It concludes with a classification of pseudocharacters (pseudo-TQFTs) and their generating functions for the category of oriented two-dimensional cobordisms in the characteristic 0 case.Contents 1 Introduction 15 2 Universal construction in monoidal and non-monoidal settings 19 3 Brauer envelopes of categories 28 4 Pseudocharacters and one-dimensional topological theories 43 5 Two-dimensional pseudo-TQFTs 60
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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 | 0.001 | 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".