The history of higher category theory has lead to a wealth of definitions, each built on differing ideas and principles. Until recently there has been very little in the way of machinery to compare them. My ongoing work, joint with Clark Barwick, is changing this.

- 4-manifold Abel Lecture algebraic K-theory axiomization calibrated geometry Cerf theory Clark Barwick cobordism hypothesis comparison problem Costello emily riehl Frobenius algebra generators and relations higher category theory homotopy category homotopy theory Mike Hopkins Morse 2-function ninja open-closed quantum field theory Stolz-Teichner symmetric monoidal bicategories topological conformal field theory topological field theories unicity Yasha Savelyev