Model category of marked objects

TL;DR

This paper introduces a new model category framework for marked objects in a given model category, with applications to simplicial sets and quasi-categories, enabling the study of limits of diagrams within these structures.

Contribution

It defines a general construction of model categories of marked objects and applies it to simplicial sets, advancing the understanding of $( abla,1)$-categories with limits.

Findings

Constructed a model category of marked objects for any functor to a model category.

Applied the construction to simplicial sets with the Joyal model structure.

Developed a model category of quasi-categories with all diagram limits.

Abstract

For every functor $\mathcal{F} : \mathcal{K} \to \mathbf{C}$, where $\mathcal{K}$ is a small category and $\mathbf{C}$ is a model category which satisfies some mild hypotheses, we define a model category $\mathbf{C}^m$ of $\mathcal{K}$-marked objects of $\mathbf{C}$. We consider an application of this construction to the category of simplicial sets with the Joyal model structure. Marked simplicial sets can be thought of as $(\infty,1)$-categories with some additional structure which depends on $\mathcal{F}$. In particular, we construct a model category of quasi-categories which have limits of all diagrams of any given shape.