Double categorical model of $(\infty,1)$-categories

TL;DR

This paper develops four model structures on double categories to model various higher categorical structures, including $( abla,1)$-categories, and provides explicit formulas for homotopy colimits, advancing the understanding of higher category theory.

Contribution

It constructs four new model structures on double categories, explicitly models $( abla,1)$-categories, and offers formulas for homotopy colimits, extending higher category theory frameworks.

Findings

Four model structures on double categories are constructed.

A formula for homotopy colimits in these models is provided.

The model for $( abla,1)$-categories is proposed to mirror classical models in homotopy theory.

Abstract

Building on work by Fiore-Pronk-Paoli, we construct four model structures on the category of double categories, each modeling one of the following: simplicial spaces, Segal spaces, $(\infty,1)$-categories, and $\infty$-groupoids. Additionally, we provide an explicit formula for computing homotopy colimits in these models using the Grothendieck construction. We expect the model of double categories for $(\infty,1)$-categories to play a similar role than that of the model of categories for spaces or $\infty$-groupoids in Grothendieck's study of the homotopy theory of spaces.