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

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, $(\mathrm{\infty },1)$-categories, and $\mathrm{\infty }$-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 $(\mathrm{\infty },1)$-categories to play a similar role than that of the model of categories for spaces or $\mathrm{\infty }$-groupoids in Grothendieck’s study of the homotopy theory of spaces.