Localizations of higher categories with applications

具有应用程序的更高类别的本地化

• 批准号: RGPIN-2015-04095
• 负责人: Pronk, Dorothea
• 金额: $ 1.02万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=662642
• 关键词: Localizations; higher; categories; 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.*** *** ********

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。 * Moerdijk和我在1995年引入了一种不同的方法来描述orbifolds，相应的orbifolds之间的映射（以及这些映射之间的映射）的概念允许我们引入orbifolds的同伦不变量（它们的形状而不是几何的特征）。它们在拓扑量子场论（TQFT）中被证明是非常有用的，TQFT是一种描述量子场的性质的理论，它只依赖于它的形状，而不依赖于它的几何形状。* 在这项研究中，我想给出另一种轨道的表示，它将阐明轨道和TQFT之间的深层联系，并使我们能够将关于轨道的结果转化为关于TQFT的结果。这方面的关键见解是在轨道褶皱之间的地图和地图集图表之间的对应关系上建立进一步的结构。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。* 作为这些新表示的进一步应用，我计划获得轨道折叠的新同伦不变量;我们还希望进一步推广轨道折叠的概念，以便我们能够使用相同的技术来研究具有局部对称性概念的其他情况。我的第二个研究方向是研究和开发一个新的弱高范畴模型。高维范畴理论在研究结构和结构之间的关系以及这些关系之间的关系时很有用，就像轨道折叠一样。有几个模型用于弱的更高类别，但它们非常技术性。我们的新模型是以这样一种方式制作的，它将更容易归纳定义我们在低维中知道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。我们希望应用这一点的领域之一是研究TQFTs中发生的干扰。 ***********