# Degeneracies in quasi-categories

## TL;DR

This paper demonstrates that semisimplicial sets with the weak Kan condition can be equipped with a simplicial structure under certain conditions, and shows the equivalence of different such structures using a combinatorial approach.

## Contribution

It introduces a new combinatorial method to establish simplicial structures on semisimplicial sets with the weak Kan condition, extending to semi-simplicial spaces.

## Key findings

- Semisimplicial sets with weak Kan condition can admit a simplicial structure.
- Different choices of simplicial structures are equivalent in the quasi-categorical sense.
- The method applies to semi-simplicial objects in various categories, including semi-simplicial spaces.

## Abstract

In this note we show that a semisimplicial set with the weak Kan condition admits a simplicial structure, provided any object allows an idempotent self-equivalence. Moreover, any two choices of simplicial structures give rise to equivalent quasi-categories. The method is purely combinatorial and extends to semisimplicial objects in other categories; in particular to semi-simplicial spaces satisfying the Segal condition (semi-Segal spaces).