Abstract:

Calaque–Scheimbauer’s work (inspired by the Baez–Dolan cobordism hypothesis, Lurie’s formulation of higher TQFTs, and the theory of higher Morita categories) develops a framework to view manifolds with corners in an $(\infty,n)$-category. One of the key technical setups involves complete Segal spaces (in the sense of Rezk), adapted to capture the structure of $(\infty,n)$-categories. This seminar series will step through the necessary background, outline Calaque–Scheimbauer’s construction, and highlight the crucial conditions—namely the Segal and “essential constancy” conditions—that guarantee a proper higher-categorical structure.

In talk 1, out of 4, we will motivate the study of $\infty$-categories, with a focus on why “higher” categorical methods (such as $(\infty,n)$-categories) are essential when dealing with extended topological quantum field theories and other modern topics in geometry/topology. We will then briefly overview the various models for $\infty$-categories—quasi-categories, Segal spaces, and complete Segal spaces—and discuss the high-level idea of how each model encodes composition up to higher homotopies. This will set the stage for the specific approach via complete Segal spaces in subsequent talks.