Localizations of higher categories with applications

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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.