Cyclic Segal Spaces

TL;DR

This survey explores the cyclic category and introduces new model structures for cyclic Segal and 2-Segal spaces, providing conceptual insights and detailed proofs in the area of higher category theory.

Contribution

It offers a comprehensive review of cyclic categories, presents a new proof of the cyclic set model structure, and develops model structures for cyclic Segal and 2-Segal spaces.

Findings

New proof of the model structure on cyclic sets

Development of model structures for cyclic Segal spaces

Introduction of model structures for cyclic 2-Segal spaces

Abstract

In this survey article, we review some conceptual approaches to the cyclic category $\Lambda$, as well as its description as a crossed simplicial group. We then give a new proof of the model structure on cyclic sets, work through the details of the generalized Reedy structure on cyclic spaces, and introduce model structures for cyclic Segal spaces and cyclic 2-Segal spaces.