Relative Calabi-Yau structures

The basic operation of oriented cobordism is to glue two oriented manifolds along a common boundary component to produce a new oriented manifold. In this talk, we discuss a generalization of this procedure to noncommutative geometry: we introduce the concept of a Calabi-Yau structure on a functor of differential graded categories which should be interpreted as an analog of an oriented manifold with boundary. As an application of the resulting theory, we show that topological Fukaya categories of surfaces give rise to a 2D TFT with values in Calabi-Yau cospans of differential graded categories.

Based on joint work in progress with Chris Brav.