On notions of compactness, object classifiers, and weak Tarski universes

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Authors

STENZEL Raffael

Year of publication 2023
Type Article in Periodical
Magazine / Source Mathematical Structures in Computer Science
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.1017/S0960129523000051
Doi http://dx.doi.org/10.1017/S0960129523000051
Keywords relative compactness; object classifiers; Tarski universes; presentable 8-categories; combinatorial model categories
Description We prove a correspondence between kappa-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively kappa-compact maps in their underlying quasi-categories for suitably large regular cardinals kappa. We thus obtain a transition result between weakly universal small fibrations in the (type-theoretic) injective Dugger-Rezk-style standard presentations of model toposes and object classifiers in Grothendieck infinity-toposes in the sense of Lurie.
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