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Record W329326931 · doi:10.1515/tmj-2015-0001

Pitts monads and a lax descent theorem

2015· article· en· W329326931 on OpenAlex
Marta Bunge

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

Bibliographic record

VenueTbilisi Mathematical Journal · 2015
Typearticle
Languageen
FieldMathematics
TopicHomotopy and Cohomology in Algebraic Topology
Canadian institutionsMcGill University
Fundersnot available
KeywordsMathematicsDescent (aeronautics)Topos theoryMorphismMonad (category theory)Bounded functionPure mathematicsDiscrete mathematicsAlgebra over a fieldFunctorMathematical analysis

Abstract

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A theorem of A.M.Pitts (1986) states that essential surjections of toposes bounded over a base topos $\mathscr{S}$ are of effective lax descent. The symmetric monad $\mathscr{M}$ on the 2-category of toposes bounded over $\mathscr{S}$ is a KZ-monad (Bunge-Carboni 1995) and the $\mathscr{M}$-maps are precisely the $\mathscr{S}$-essential geometric morphisms (Bunge-Funk 2006). These last two results led me to conjecture1 and then prove2 the general lax descent theorem that is the subject matter of this paper. By a ‘Pitts KZ-monad’ on a 2-category $\mathscr{K}$ it is meant here a locally fully faithful equivariant KZ-monad $\mathscr{M}$ on $\mathscr{K}$ that is required to satisfy an analogue of Pitts' theorem on bicomma squares along essential geometric morphisms. The main result of this paper states that, for a Pitts KZ-monad $\mathscr{M}$ on a 2-category $\mathscr{K}$ (‘of spaces’), every surjective $\mathscr{M}$-map is of effective lax descent. There is a dual version of this theorem for a Pitts co-KZ-monad $\mathscr{N}$. These theorems have (known and new) consequences regarding (lax) descent for morphisms of toposes and locales.

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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)0.002
metaresearch head score (Gemma)0.003
Version: codex-gemma-dda1882f352aValidation 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: Empirical
Teacher disagreement score0.042
Threshold uncertainty score0.721

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0020.003
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.001
Insufficient payload (model declined to judge)0.0010.000

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.

Opus teacher head0.068
GPT teacher head0.332
Teacher spread0.264 · 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