Categorical algebra in analysis, geometry, and topology

很多数学都是研究空间和数量，以及它们之间的关系。这里的空间概念不仅包括我们周围熟悉的三维物理空间，还包括其中各种形状和图形的配置，以及更高维度的空间，以及从这些起点抽象出来的各种空间数学概念。这里的量的概念不仅包括熟悉的计数数、有理数和实数，而且还包括在空间中随点变化的数值，如地球表面的温度，以及在空间中分布的量，如通过地球大气层分布的某些气体的量。函数、度量和分布的数学概念以及各种其他相关概念被设计用来在数学上表示这样的变量和分布量。空间有各种微妙不同的数学概念，如拓扑空间、流形和簇的概念，每一种概念都由其自己的数学分支研究，如拓扑学、微分几何和代数几何。相应地，有许多微妙的不同的函数和分布概念与不同的空间概念相关联，这些概念最终以意想不到的方式被应用，例如在概率和统计学研究中描述实验可能结果的空间的可能性分布。这项拟议的研究采用了范畴理论的方法，这是对数学中结构的一般研究，以识别和理解所有这些微妙不同的空间和数量概念所共有的共同结构特征，并发展一套范畴理论框架、公理和结果体系，以促进对产生和表征变量和分布量的结构的新的和统一的理解。这将有助于使其他数学分支的实践者更容易接触到不同数学分支的概念和方法，通过这些结构研究获得的见解使数学中新的有效方法成为可能。此外，这些结构研究使他们的数学见解能够更容易地转移到应用领域，例如在概率编程语言的设计中，并且它们支持为此目的有效地改变和适应这些想法。

Much of mathematics is concerned with the study of space and quantity, and the relation of each to the other. The notion of space here comprises not only the familiar three-dimensional physical space that surrounds us but also the configurations of various shapes and figures therein, as well as higher-dimensional spaces, and various mathematical concepts of space that abstract from these beginnings. The notion of quantity here includes not only the familiar counting numbers, rational, and real numbers, but also numerical quantities that vary from point to point over a space, such as the temperature at the surface of the earth, and also quantities that are distributed through space, such as the quantity of certain gas distributed through the earth's atmosphere. The mathematical notions of function, measure, and distribution, as well as various other related notions, were devised to represent mathematically such variable and distributed quantities. There are various subtly different mathematical notions of space, such as the notions of topological space, manifold, and variety, each studied by its own branch of mathematics, such as topology, differential geometry, and algebraic geometry. Correspondingly, there are numerous subtly different notions of function and distribution associated with different notions of space, and these notions end up being applied in unexpected ways, e.g. to describe the distribution of likelihood across a space of possible outcomes of an experiment, as in the study of probability and statistics. The proposed research employs the methods of category theory, a general study of structure in mathematics, to identify and understand the common structural characteristics shared by all these subtly different notions of space and quantity, and to develop a system of category-theoretic frameworks, axiomatics, and results that foster a new and unified understanding of structures that generate and characterize variable and distributed quantities in general.  This will serve to make the notions and methods of each of various branches of mathematics more accessible to the practitioners of the others, and the insights gained by these structural studies enable new and effective methods in mathematics.  Further, these structural studies allow mathematical insights to be more readily transferred to application domains, e.g. in the design of probabilistic programming languages, and they support the effective variation and adaptation of these ideas to that end.