Higher dimensional categories

数学的标准基础由集合论和Zermelo-Fraenkel公理提供。范畴理论最初是作为一种工具出现的，用来组织某些数学分支，如代数、拓扑学和代数几何。从20世纪60年代开始，S逐渐认识到范畴理论可能会取代集合论成为整个数学的基础。在这方面的决定性发现是F·W·劳维尔和M·蒂尔尼在1970年提出的基本拓扑概念。拓扑学是一个可以在其中发展数学的领域。拓扑语的对象被认为是抽象的集合，它们通过作为抽象函数的态射和由拓扑语的公理所提供的结构相互联系。TOPO是有限的，因为通常的大类别本身不是它的对象。人们需要另一个宇宙，然而，它不再是一个范畴，而是一个二维范畴，以将范畴容纳为对象。对更高维度类别的需求将不会停止，宇宙必须被视为由所有n-类别组成，对于不同的类别。新基础理论的另一个方面是认识到，宇宙的实体将根据它们的类型而具有不同的身份概念。这一事实导致了所谓的弱n-范畴，它的精确确定是高维范畴理论目前的主要任务。HDC对于各种数学目的也很重要，在理论物理中也是如此。虽然鼓掌者的工作集中在HDC的基础方面，而不是更多的数学方面，但他使用了所有可用的范畴理论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.