Localizations of higher categories with applications

流形是光滑的物体，就像一个内管或一个球的表面，尽管它们不需要是二维的，并且自世纪以来在数学中得到了广泛的研究。流形的光滑性是精确的，因为局部的物体看起来就像欧氏n-空间的平面。1956年，一种叫做orbifolds的流形被引入。Orbifolds可能有一些尖锐的点。一个例子的orbifold将是一个冰淇淋锥，它是从一个圆形片获得通过制作一个2倍或3倍的覆盖，并具有一个尖锐的锥点的地方，该中心的圆。我们利用圆形区域的2重或3重对称性来制作这个物体。一般来说，轨道褶皱可以局部地描述为具有有限对称性的光滑物体，这会产生一些我们称之为奇点的尖锐特征。所有这些局部描述的集合，以及它们如何组合在一起的一些信息，给出了一个眶褶的地图集。

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.