Infinitary generalizations of Deligne's completeness theorem

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F20%3A00118489" target="_blank" >RIV/00216224:14310/20:00118489 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1017/jsl.2020.27" target="_blank" >https://doi.org/10.1017/jsl.2020.27</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1017/jsl.2020.27" target="_blank" >10.1017/jsl.2020.27</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Infinitary generalizations of Deligne's completeness theorem

Popis výsledku v původním jazyce

Given a regular cardinal kappa such that kappa(&lt;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.

Název v anglickém jazyce

Infinitary generalizations of Deligne's completeness theorem

Popis výsledku anglicky

Given a regular cardinal kappa such that kappa(&lt;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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Symbolic Logic

ISSN

0022-4812

e-ISSN

—

Svazek periodika

85

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

16

Strana od-do

1147-1162

Kód UT WoS článku

000628900500012

EID výsledku v databázi Scopus

2-s2.0-85102715919