Localizations of higher categories with applications

Manifolds are smooth objects like an inner tube or the surface of a ball, although they don't need to be 2-dimensional, and have been studied extensively in mathematics since the 19th century. The smoothness of a manifold is made precise by saying that locally the object looks just like the plane of like Euclidean n-space. In 1956 a generalization of manifolds called orbifolds was introduced. Orbifolds may have some sharp points. An example of an orbifold would be an ice cream cone which is obtained from a circular sheet by making a 2-fold or 3-fold covering and which has a sharp cone point at the place where the center of the circle was. We used the 2- or 3-fold symmetry of the circular area to make this object. In general, an orbifold can locally be described as a smooth object with a finite amount of symmetry which gives rise to some sharp features which we call singularities. The collection of all these local descriptions together with some information on how they fit together gives an atlas for the orbifold.

Moerdijk and I introduced a different way of describing orbifolds in 1995 and the corresponding notion of map between orbifolds (and of maps between these maps) has allowed us to introduce homotopy invariants for orbifolds (a characteristic of their shape rather than their geometry). They have proven very useful in topological quantum field theory (TQFT), a theory that describes the properties of a quantum field that only depend on its shape, not on its geometry.

In this research I want to give yet another representation of orbifolds which will clarify the deep connection between orbifolds and TQFTs and will enable us to translate results about orbifolds into results about TQFTs. The key insight for this is to put further structure on the maps between orbifolds and on the correspondences between atlas charts. This can be done in several ways, all based on existing constructions in (weak) higher category theory.

As a further application of these new representations I plan to obtain new homotopy invariants for orbifolds; we also hope to further generalize the notion of orbifold so that we may be able to use the same techniques to study other situation with a concept of local symmetry.

My second research direction is to study and develop a new model for weak higher categories. Higher dimensional category theory is useful in studying situations where we have structures and relationships between structures and then again relationships between those relationships, as is the case for orbifolds. There are a couple of models for weak higher categories, but they are very technical to work with. Our new model is made in such a way that it will be easier to inductively define higher dimensional analogues of constructions that we know and understand in low dimensions. Our model differs from others in that we introduce the weakness in a non-standard way. One of the areas where we hope to apply this is in the study of the duals occurring in TQFTs.