On notions of compactness, object classifiers and weak Tarski universes

TL;DR

This paper establishes a correspondence between small fibrations in certain model categories and compact maps in their underlying quasi-categories, bridging concepts in model toposes and Grothendieck $$-toposes.

Contribution

It provides a transition result linking weakly universal small fibrations in model toposes with object classifiers in Grothendieck $$-toposes, enhancing the understanding of their relationship.

Findings

Established a correspondence between $$-small fibrations and relatively $$-compact maps.

Bridged concepts between model toposes and Grothendieck $$-toposes.

Provided a transition result for large regular cardinals.

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.