Notes on the Joyal model structure

TL;DR

This paper introduces a new construction of the Joyal model structure on simplicial sets, offering a simplified characterization of fibrations and inner anodyne maps, and extends the structure to simplicial sets with fixed zero-simplices.

Contribution

It provides a novel construction and characterization of the Joyal model structure, simplifying understanding of fibrations and inner anodyne maps, and extends the structure to fixed zero-simplices.

Findings

New construction of the Joyal model structure

Characterization of fibrations and inner anodyne maps

Extension to simplicial sets with fixed zero-simplices

Abstract

We give a new construction of the Joyal model structure on the category of simplicial sets, and we provide a simple characterization of the fibrations in it. We characterize the inner anodyne maps in terms of categorical equivalences and use this characterization to establish the inner model structure on the category of simplicial sets whose set of zero-simplices is equal to a fixed set $O$.