Monoidal categories, homological algebra, and applications

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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.