Dualizability in Low-Dimensional Higher Category Theory

摘要

These lecture notes form an expanded account of a course given at the SummerSchool on Topology and Field Theories held at the Center for Mathematics at theUniversity of Notre Dame, Indiana during the Summer of 2012. A similar lectureseries was given in Hamburg in January 2013. The lecture notes are divided intotwo parts. The first part, consisting of the bulk of these notes, provides an expositoryaccount of the author's joint work with Christopher Douglas and Noah Snyder ondualizability in low-dimensional higher categories and the connection tolow-dimensional topology. The cobordism hypothesis provides bridge betweentopology and algebra, establishing important connections between these twofields. One example of this is the prediction that the $n$-groupoid ofso-called `fully-dualizable' objects in any symmetric monoidal $n$-categoryinherits an O(n)-action. However the proof of the cobordism hypothesis outlinedby Lurie is elaborate and inductive. Many consequences of the cobordismhypothesis, such as the precise form of this O(n)-action, remain mysterious.The aim of these lectures is to explain how this O(n)-action emerges in a rangeof low category numbers ($n \leq 3$). The second part of these lecture notes focuses on the author's joint workwith Clark Barwick on the Unicity Theorem, as presented in arXiv:1112.0040.This theorem and the accompanying machinery provide an axiomatization of thetheory of $(\infty,n)$-categories and several tools for verifying these axioms.The aim of this portion of the lectures is to provide an introduction to thismaterial.