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Record W2152171487 · doi:10.4171/dm/392

Coherence for weak units

2013· preprint· en· W2152171487 on OpenAlexaff
André Joyal, Joachim Kock

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.

How this classification was reachedexpand

Full frame machine prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.002
metaresearch head score (Gemma)0.003
Version: metacan-v3-hybrid-931329e0061cValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: none
Teacher disagreement score0.006
Threshold uncertainty score0.020

Distilled classifier scores by category (both heads)

CategoryCodexGemma
Metaresearch0.0020.003
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.001
Bibliometrics0.0020.001
Science and technology studies0.0020.005
Scholarly communication0.0030.007
Open science0.0010.005
Research integrity0.0010.002
Insufficient payload (model declined to judge)0.0060.001

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

Classification

machine, unvalidated

Machine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.

The models applied no category: nothing in the taxonomy fit this work.
Study designTheoretical or conceptual
Domainnot available
GenreEmpirical

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".

Quick stats

Citations8
Published2013
Admission routes1
Has abstractyes

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