Localizations of higher categories with applications

流形是光滑的物体，像内管或球的表面，尽管它们不需要是二维的，自19世纪以来在数学中得到了广泛的研究。流形的光滑性是通过说对象在局部看起来就像欧几里得n空间的平面而精确的。1956年，流形的一个推广被引入到流形中。奥比诺兹可能有一些尖锐的观点。冰激凌圆锥体就是冰激凌圆锥体的一个例子。冰激凌圆锥体是由一个圆形薄片制成的，由两层或三层覆盖而成，在圆心所在的位置有一个尖锐的圆锥点。我们利用圆形区域的2或3重对称性来制作这个物体。在一般情况下，奥比福尔可以局部地描述为具有有限对称量的光滑对象，这产生了一些尖锐的特征，我们称之为奇点。所有这些地方描述的集合，加上一些关于它们如何组合在一起的信息，形成了一本奥比福尔德地图集。*Moerdijk和我在1995年引入了一种不同的方法来描述ordioles，相应的orbiorold之间的映射(以及这些映射之间的映射)的概念使我们能够为orbioles引入同伦不变量(一个关于它们的形状的特征，而不是它们的几何特征)。它们在拓扑量子场论(TQFT)中被证明是非常有用的，TQFT是一种描述只依赖于其形状而不是其几何的量子场的性质的理论。在本研究中，我想给出另一种奥布洛德的表示，它将阐明奥布洛德和TQFT之间的深层次联系，并使我们能够将关于奥布朗德的结果转化为关于TQFT的结果。这方面的关键洞察力是在奥比诺德之间的地图和地图集图表之间的对应关系上加入更多的结构。这可以通过几种方式来实现，所有这些都基于(弱)高级范畴理论中的现有结构。*作为这些新的表示的进一步应用，我计划得到新的奥比霍尔德的同伦不变量；我们还希望进一步推广奥比霍尔德的概念，这样我们就可以用同样的技巧来研究具有局部对称性的其他情形。*我的第二个研究方向是研究和发展弱高范畴的一个新模型。高维范畴理论在研究我们有结构和结构之间的关系，然后再研究这些关系之间的关系的情况下很有用，就像奥比诺德的情况一样。有几个模型适用于较弱的较高类别，但它们使用起来非常技术性。我们的新模型是这样一种方式，它将更容易归纳定义我们在低维中已知和理解的结构的高维类似物。我们的模型与其他模型的不同之处在于，我们以非标准的方式引入了弱点。我们希望将其应用于对TQFT中出现的对偶的研究。

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.*** *** ********