On notions of compactness, object classifiers and weak Tarski universes

Abstract

We prove a correspondence between κ-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relative κ-compact maps in their underlying quasi-categories for suitably large inaccessible cardinals κ. We thus obtain a transition result between weakly universal small fibrations in the injective Dugger-Rezk-style standard presentations of model toposes and object classifiers in Grothendieck ∞-toposes in the sense of Lurie.especially, to Mike Shulman for sharing his note on the presentation of small (∞, 1)-categories as localizations of inverse posets, which provided a crucial step to the proof of Theorem 3.16. Most of the work for this paper was carried out as part of the author's PhD thesis, supported by a Faculty Award of the University of Leeds 110 Anniversary Research Scholarship. Replacing simplicial categories with direct posetsIn the following, simplicial categories -that is simplicially enriched categories -will be denoted by bold faced letters C and ordinary categories will be distinguished by blackboard letters C. S denotes the (simplicial) category of simplicial sets. By a simplicial presheaf over C we mean a simplicially enriched presheaf X : C op → S. Simplicial functors and simplicial natural transformations form a simplicial category sPsh(C) whose underlying ordinary category also will be denoted by sPsh(C).Mike Shulman noted in [20, Lemma 0.2] that every quasi-category can be presented by the localization of a direct -in other words, well founded -poset. 1 Since the note is unpublished, in this section we present a slight variation of his observation (with a different proof) and discuss the resulting presentations of associated presheaf ∞-categories. Although the following sections only will require the fact that every quasi-category can be presented by the localization of an Eilenberg-Zilber Reedy category, proving the stronger condition of posetality only requires about as much work as the Eilenberg-Zilber Reedy condition itself.