Complete Segal Objects

TL;DR

This paper introduces complete Segal objects as internal models of higher categories, explores their properties via Cartesian fibrations, and applies them to define univalence in non-presentable categories, extending existing theories.

Contribution

It defines complete Segal objects, studies their limits and adjunctions, and generalizes univalence to non-presentable locally Cartesian closed categories.

Findings

Complete Segal objects serve as internal higher category models.

Representation of limits and adjunctions for these objects is established.

Univalence is defined in non-presentable categories, broadening previous frameworks.

Abstract

We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.