Monoidal categories, homological algebra, and applications

拟议的研究计划侧重于使用分类机器来解决不同数学领域的问题。范畴理论是由Eilenberg和Mac Lane发明的，他们在1940年代的S中引入了范畴和函子的概念，用于代数拓扑学，一种关于形状的代数研究。我们的目标是理解保持数学结构的过程，因此由此产生的理论是如此普遍，以至于它基本上涵盖了所有的数学。有人可能会说，数学本身关注的是问题的抽象形式化，从而导致问题的解决。也就是说，为了解决一个问题，我们试图通过定义具有特定结构的对象来理解它，这反过来又通过坚持关系保持这种结构来限制它们之间的关系。这个对象和它们之间的关系的概念一旦形式化，就是一个范畴的定义。这一观点的一个直接结果是出现了关系高于对象的首要地位。它是一个伟大的统一工具。范畴理论的出现与现代同调代数的发展紧密交织在一起，现代同调代数的应用起源于代数拓扑学，但后来几乎涵盖了所有的数学；它研究与函子范畴概念自然产生的额外结构。研究计划中不是范畴的方面是同调代数类型的，有些是相互交织的。我们使用的一个重要概念是范畴中的么半积。在许多方面，这是一个人在学校早期遇到的数字的乘积的泛化。这个数字的乘积就是所谓的可交换，即3乘以4等于4乘以3。一般的么半群乘积不需要是可交换的，也不需要对对象A乘以对象B与对象B之间的关系进行任何限制。在所提出的研究方案中，我们可以根据交换性限制的程度来区分不同的应用领域：从没有到最强，其中后者以这样的方式识别对象A×对象B与其相反的对象，其逆与对象B与其相反的对象A的识别一致。该提案考察了应用于物理学、数论、非对易几何和k图的问题。值得注意的是，在上述列表中，应用程序之间存在应用程序。该项目的长期目标是将其中的考虑从类别扩展到无限类别。后者在调查过程中变得必要；它们将关系的层次永远向上延伸，即我们现在有对象、对象之间的关系、对象之间的关系以及无限。在某种意义上，它是范畴理论和同调代数的统一概念。

The proposed research program focuses on the use of categorical machinery in addressing questions in diverse mathematical areas. Category theory was invented by Eilenberg and Mac Lane who introduced the concepts of categories and functors in the 1940's for use in algebraic topology, an algebraic study of shapes. The goal was to understand the processes that preserve mathematical structure and as such the resulting theory is so general that it essentially encompasses all of mathematics. One may say that mathematics concerns itself with the abstract formalization of problems that results in their solutions. Namely, to solve a problem we try to understand it by defining objects with a specified structure, which in turn places restrictions on the relationships between them, by insisting that the relationships preserve this structure. This concept of objects and relationships between them, once itself formalized, is a definition of a category. One immediate outcome of this point of view is the emergence of the primacy of the relationships over the objects. It is a great unification tool. The emergence of category theory was closely entwined with the development of modern homological algebra whose applications originated in algebraic topology but have since encompassed almost all of mathematics; it studies additional structures that naturally arise with the categorical notion of a functor. The aspects of the research program that are not categorical are of the homological algebra type, some are intertwined. An important notion that we use is that of the monoidal product in categories. It is a generalization, in many ways, of the product of numbers that one encounters early on in school. That product of numbers is what is known as commutative, i.e., 3 times 4 is the same as 4 times 3. General monoidal products need not be commutative, nor any restrictions on the relationships between object A times object B and the opposite need be required. In the proposed research program we can distinguish different areas of application by the degree of the commutativity restrictions: from none to the strongest possible, where the latter identifies object A times object B with the opposite in such a way that its inverse coincides with the identification of object B times object A with its opposite. The proposal examines matters with applications to physics, number theory, noncommutative geometry, and k-graphs. It is worth noting that in the above list, there are applications between applications. The long term goal of the project is to extend the considerations therein from categories to infinity categories. The latter become necessary in the course of investigations; they extend the hierarchy of relationships forever upward, i.e., we now have objects, relationships between objects, relationships between relationships between objects, and ad infinitum. In a sense it is a notion that unites both category theory and homological algebra.