Abstract

Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models, like Bénabou’s bicategories, tricategories following Gurski and the models of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, including those of Tamsamani and Paoli, along with quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell, Nikolaus and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of $(\infty, n)$-category for general $n$.In this talk, I will explore current work in the problem of taking homotopy bicategories of non-algebraic $(\infty, 2)$-categories, with particular focus on the model of complete 2-fold Segal spaces, including a construction of my own. If time permits, I will discuss potential applications of this problem to connecting extended TQF Ts in the world of (oo, 2)-categories and in bicategories.