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Record W2152171487 · doi:10.4171/dm/392

Coherence for weak units

2013· preprint· en· W2152171487 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueDocumenta Mathematica · 2013
Typepreprint
Languageen
FieldMathematics
TopicHomotopy and Cohomology in Algebraic Topology
Canadian institutionsUniversité du Québec à Montréal
Fundersnot available
KeywordsMathematicsMorphismAxiomEquivalence (formal languages)Coherence (philosophical gambling strategy)Unit (ring theory)Contractible spaceWeak equivalencePure mathematicsDiscrete mathematicsCombinatoricsGeometryPhysicsQuantum mechanics

Abstract

fetched live from OpenAlex

We define weak units in a semi-monoidal 2-category \mathcal C as cancellable pseudo-idempotents: they are pairs (I,\alpha) where I is an object such that tensoring with I from either side constitutes a biequivalence of \mathcal C , and \alpha: I \otimes I \to I is an equivalence in \mathcal C . We show that this notion of weak unit has coherence built in: Theorem A: \alpha has a canonical associator 2-cell, which automatically satisfies the pentagon equation. Theorem B: every morphism of weak units is automatically compatible with those associators. Theorem C: the 2-category of weak units is contractible if non-empty. Finally we show (Theorem E) that the notion of weak unit is equivalent to the notion obtained from the definition of tricategory: \alpha alone induces the whole family of left and right maps (indexed by the objects), as well as the whole family of Kelly 2-cells (one for each pair of objects), satisfying the relevant coherence axioms.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.001
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesMeta-epidemiology (narrow), Insufficient payload (model declined to judge)
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.219
Threshold uncertainty score1.000

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.001
Meta-epidemiology (narrow)0.0010.001
Meta-epidemiology (broad)0.0010.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0010.001
Research integrity0.0010.001
Insufficient payload (model declined to judge)0.0050.001

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.

Opus teacher head0.073
GPT teacher head0.352
Teacher spread0.278 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it