Towards an n-category of cobordisms

Abstract

We discuss an approach to constructing a weak n-category of cobordisms. First we present a generalisation of Trimble's definition of n-category which seems most appropriate for this construction; in this definition composition is parametrised by a contractible operad. Then we show how to use this definition to define the n-category nCob, whose k-cells are k-cobordisms, possibly with corners. We follow Baez and Langford in using "manifolds embedded in cubes" rather than general manifolds. We make the construction for 1-manifolds embedded in 2- and 3-cubes. For general dimensions k and n we indicate what the construction should be. © Eugenia Cheng and Nick Gurski, 2007.