Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property which characterizes descent and hence classifying maps in (∞, 1)-categories. Completeness is a property of Segal spaces introduced by Charles Rezk which characterizes those Segal spaces that are (∞, 1)-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study ramifications of this analogy.More precisely, we generalize univalence of fibrations and Rezk-completeness of Segal spaces to two properties of Segal objects in well behaved type theoretic model categories, and give a characterization of these two generalized properties in terms of each other. Furthermore, in this context we give two definitions of Dwyer-Kan equivalences between Segal objects and between fibrations, respectively, and obtain a functor of associated relative categories which translates univalent completion in the sense of van den Berg and Moerdijk to Rezk-completion of associated Segal objects. Motivated by these correspondences, we will show that univalent fibrations are exactly the DK-local fibrations whenever univalent completion exists.

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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.

We introduce an ∞-category of geometric ∞-categories, whose objects are defined by dropping the effectivity condition on colimits in Anel and Joyal's definition of ∞-logos. We propose a theory of higher geometric sheaves on geometric ∞categories C, which will be shown to differ from the ordinary geometric sheaf theory on C by a cotopological fragment. We prove that this fragment is crucial: for instance every ∞-topos is the theory of higher geometric sheaves over itself, but the corresponding cotopological localization of its ordinary geometric sheaf theory is generally non-trivial. The notion of higher geometric sheaves over geometric ∞categories hence faithfully generalizes Lurie's definition of sheaves over ∞-toposes.We define this class of sheaf theories by way of an adaption of Anel and Leena Subramaniam's ∆-modulators. The according sheaves are characterized by a limit preservation property that is generally not captured by the classical sheaf condition as known from ordinary sheaf theory over topological spaces (or over geometric categories more generally). The latter arises as a special case instead. To motivate this class of limits, we first introduce and discuss simpler but analogous sheaf theories for extensive and regular ∞-categories.