Introduction to Complete Segal Spaces

TL;DR

This paper provides an accessible overview of complete Segal spaces, focusing on their categorical properties, functoriality, limits, and adjunctions, with an emphasis on examples and intuition.

Contribution

It offers an expository introduction to complete Segal spaces, clarifying how categorical notions are interpreted and behave within this framework.

Findings

Clarification of functoriality via fibrations

Definition of limits and adjunctions in complete Segal spaces

Emphasis on examples and intuitive understanding

Abstract

The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces via fibrations. Then we use it to define limits and adjunctions in a complete Segal space. This is mostly expository and the focus is on examples and intuition.