Monoidal categories, homological algebra, and applications

拟议的研究计划的重点是使用分类机器在不同的数学领域解决问题。范畴理论是由艾伦伯格和麦克莱恩发明的，他们在20世纪40年代引入了范畴和函子的概念，用于代数拓扑学，一种对形状的代数研究。目标是理解保持数学结构的过程，因此产生的理论是如此普遍，以至于它基本上包含了所有的数学。有人可能会说，数学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，这反过来又对它们之间的关系施加了限制，通过坚持这些关系保持这种结构。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关系优先于对象的出现。这是一个伟大的统一工具。范畴论的出现与现代同调代数的发展密切相关，现代同调代数的应用起源于代数拓扑学，但此后几乎涵盖了所有数学;范畴论研究的是函子范畴概念自然产生的额外结构。研究纲领中非范畴的方面是同调代数类型的，有些是交织在一起的。 我们使用的一个重要概念是范畴中的monoidal积。在很多方面，它是一个在学校早期遇到的数字乘积的概括。这个数的乘积就是所谓的交换数，即，3乘以4与4乘以3相同。一般的么半群积不需要是交换的，也不需要对物体A乘以物体B与相反物体之间的关系有任何限制。在所提出的研究计划中，我们可以根据交换性限制的程度来区分不同的应用领域：从没有到最强的可能，后者将对象A乘以对象B与其对立面进行识别，使得其对立面与对象B乘以对象A与其对立面的识别相一致。该提案探讨了物理学，数论，非交换几何和k-图的应用问题。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因素从类别扩展到无限类别。后者在调查过程中成为必要的;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，以及无限的关系。在某种意义上，它是一个统一范畴论和同调代数的概念。

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.