Seminars and short courses

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 more homotopy-theoretic ones like 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. Developing comparisons between these forms of higher category has relevance to topological quantum field theories in relating the work done on fully extended TQFTs using homotopy theoretic models of higher category, such as Lurie's proof-sketch of the Cobordism Hypothesis conducted using n-fold Segal spaces, and the large body of work on extended TQFTs using algebraic models of higher category, such as symmetric monoidal bicategories.

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. One might see this as an extension of the fundamental n-groupoid of a homotopy type, a statement I will make precise. In this talk, I will explore current work in the problem of taking homotopy bicategories of non-algebraic $(\infty, 2)$-categories, including a construction of my own. If time permits, I will discuss the connections of this problem to topological quantum field theories.