A Double Categorical Model of (∞,1)-Categories

In this talk, I will present recent joint work with Lyne Moser (arXiv:2412.15715), in which we prove that the category of double categories can be equipped with a model category structure which is related, via a zig-zag of Quillen equivalences, to the Rezk model structure on simplicial spaces. In particular, with this model structure, the category of double categories serves as a model for the homotopy theory of (∞,1)-categories.
Along with sketching the proof of this theorem, I will highlight some notable features of this model of (∞,1)-categories, such as the ability to compute homotopy colimits in a very explicit and straightforward manner. Time permitting, I will also discuss some conjectural applications of this result to future projects.