Higher categories and comparisons of different models

Abstract: 

Ordinary category theory provides a formal language to explain similar phenomena that happen in various contexts, enabling the transfer of ideas from one area of study to another. For many relevant categorical structures that occur naturally (e.g. most flavors of cobordism categories), the relations expected to hold in a category, such as associativity of composition, are not satisfied exactly but rather up to a "higher isomorphism". To accommodate these examples, it is appropriate to relax the conditions that define a category and work with a "higher category" instead. The defining principles of a higher category leave a lot of freedom of interpretation and can be implemented by many models, each with their own advantages and disadvantages depending on the purpose. I will give an overview of how one should think about a higher categorical structure, mentioning the role they play in different situations, and I will discuss some results and work in progress in the context of comparing different models of higher categories.