J 2020

Infinitary generalizations of Deligne's completeness theorem

ESPINDOLA ZAMORA, Christian Rubén

Basic information

Original name

Infinitary generalizations of Deligne's completeness theorem

Authors

ESPINDOLA ZAMORA, Christian Rubén (724 Spain, guarantor, belonging to the institution)

Edition

Journal of Symbolic Logic, Cambridge, Cambridge University Press, 2020, 0022-4812

Other information

Language

English

Type of outcome

Article in a journal

Field of Study

10101 Pure mathematics

Country of publisher

United Kingdom of Great Britain and Northern Ireland

Confidentiality degree

is not subject to a state or trade secret

References:

Impact factor

Impact factor: 0.512

RIV identification code

RIV/00216224:14310/20:00118489

Organization unit

Faculty of Science

DOI

UT WoS

000628900500012

Keywords in English

classifying topos; infinitary logics; completeness theorems; sheaf models

Tags

Tags

International impact, Reviewed
Changed: 9/4/2021 17:25, Mgr. Marie Novosadová Šípková, DiS.

Abstract

V originále

Given a regular cardinal kappa such that kappa(<kappa) = kappa (or any regular if the Generalized Continuum Hypothesis holds), we study a class of toposes with enough points, the kappa-separable toposes. These are equivalent to sheaf toposes over a site with kappa-small limits that has at most kappa many objects and morphisms, the (basis for the) topology being generated by at most. many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough kappa-points, that is, points whose inverse image preserve all kappa-small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when kappa = omega, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call kappa-geometric, where conjunctions of less than. formulas and existential quantification on less than. many variables is allowed. We prove that kappa-geometric theories have kappa-classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to kappa-geometric morphisms (geometric morphisms the inverse image of which preserves all kappa-small limits) into that topos. Moreover, we prove that kappa-separable toposes occur as the kappa-classifying toposes of kappa-geometric theories of at most. many axioms in canonical form, and that every such kappa-classifying topos is kappa-separable. Finally, we consider the case when. is weakly compact and study the kappa-classifying topos of a kappa-coherent theory (with at most. many axioms), that is, a theory where only disjunction of less than. formulas are allowed, obtaining a version of Deligne's theorem for.-coherent toposes from which we can derive, among other things, Karp's completeness theorem for infinitary classical logic.