AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN L-infinity,L-kappa

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Authors

HENRY Simon Bruno

Year of publication 2019
Type Article in Periodical
Magazine / Source Journal of Symbolic Logic
MU Faculty or unit

Faculty of Science

Citation
Web https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/an-abstract-elementary-class-nonaxiomatizable-in-linfty-kappa/87EC4FD7FED3B22F244271585EAD99B8
Doi http://dx.doi.org/10.1017/jsl.2019.25
Keywords toposes; points of toposes; Scott topology; abstract elementary classes
Description We show that for any uncountable cardinal lambda, the category of sets of cardinality at least lambda and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a L-infinity,L-omega-theory. More generally we show that for any regular cardinal kappa < lambda it is neither the category of kappa-points of a kappa-topos, in particular, nor the category of models of a L-infinity,L-kappa-theory. The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least lambda and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.
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