Higher dimensional categories

数学的标准基础是由集合论提供的，具有Zermelo-Fraenkel公理化。范畴论最初是作为一种工具出现的，组织数学的某些分支，如代数，拓扑和代数几何。从20世纪60年代开始，人们越来越认识到范畴论可能取代集合论作为整个数学的基础。这方面的决定性发现是F. W. Lawvere和M. 1970年的蒂尔尼。一个拓扑是一个范畴，作为一个宇宙，数学可以在其中发展。topos的对象被视为抽象集合;它们通过态射相互关联，态射是抽象函数，并且通过topos的公理可以得到的构造相互关联。一个topos是有限的，因为通常的大范畴本身并不是它的对象，人们需要另一个宇宙来容纳作为对象的范畴，然而，这个宇宙不再是一个范畴，而是一个二维范畴。对更高维范畴的需求不会停止，宇宙必须被认为是由所有的n-范畴组成的，对于不同的n。新基础理论的另一个方面是认识到宇宙的实体将根据其类型而具有不同的身份概念。这一事实产生了所谓的弱n-范畴，其精确确定是高维范畴理论（HDC）目前的主要任务。HDc对于各种数学目的和理论物理也很重要。虽然applcant的工作集中在HDC的基础方面，而不是更多的数学方面，但他使用了范畴论的所有可用技术来澄清和验证他的方法。

The standard foundations of mathematics are provided by Set Theory, with the Zermelo-Fraenkel axiomatization. Category Theory initially arose as a tool to organize certain branches of mathematics such as Algebra, Topology and Algebraic Geometry. Starting in the 1960's, the realization has been growing that Category Theory may replace Set Theory as the foundations of mathematics as a whole. The decisive discovery in this regard was that of the concept of elementary topos by F. W. Lawvere and M. Tierney in 1970. A topos is a category that serves as a universe in which mathematics can be developed. The objects of the topos are regarded as abstract sets; they are related to each other by the morphisms, which are abstract functions, and by the constructions that are available by the axioms of a topos. A topos is limited in that the usual large categories themselves are not objects of it. One needs another universe, which, however, is no longer a category, but a 2-dimensional category, to accomodate categories as objects. The need for ever higher dimensional categories will not stop, and the universe has to be taken as consisting of all the  n-categories, for various  n. Another aspect of the new foundational theory is the realization that entities of the universe will have varying concepts of identity, depending on their types. This fact gives rise to the so-called weak n-categories whose precise determination is the main present task of the theory of Higher Dimensional Categories (HDC's). HDc's are also important for varios mathematical purposes, and also in Theoretical Physics. Although the applcant's work is focused on the foundational aspects of HDC's rather than the more mathematical ones, he uses all the available techniques of Category Theory to clarify and validate his approach.