Mathematics 690STJ - Intro/Category Theory & Higher

All mathematical objects of a given type can often be organized into a category. Category theory offers a framework to examine how a given category of mathematical objects interacts with those of a different nature (like topological spaces and groups), and to formalize constructions that map one mathematical object to another. Algebraic topology, for example, illustrates numerous instances of this concept, where algebraic invariants are derived from topological spaces using appropriate functorial constructions. Nevertheless, it has become evident in various modern mathematical fields that many mathematical objects of interest cannot be organized into an ordinary category, or that certain constructions of interest do not behave as ordinary functors, hindering the application of standard categorical language. This issue can often be overcome by substituting standard category notions with more complex higher versions, such as the notion of a model category and of an 8-category. The course will cover fundamental concepts of standard category theory, and then delve into model category and 8-category theory.

Students should have some exposure to and familiarity with graduate level abstract mathematics, such as the theory of groups, rings and topological spaces. Having taken the graduate topology sequence is recommended but not strictly required.