Model category of marked objects

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.

13 pages