On notions of compactness, object classifiers and weak Tarski universes

Abstract

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 $\infty$-toposes in the sense of Lurie.

arXiv commentFinal version accepted for publication in MSCS. Deleted the condition of inaccessibility of the cardinals in the results as it turned out be redundant, and fixed typos