A survey of definitions of \(n\)-category (Q2768440)

The last five years or so have seen many answers to the question of what should be the best weakening of the notion of \(n\)-category. Each answer has been published with a substantial amount of auxiliary machinery. The goal of the author here is to isolate ten of these definitions, making each one precise and fairly self contained, in a two-page spread. In a further two pages, the way categories and bicategories fit the definition is explained. The reviewer is particularly attracted to the definition the author calls L' which appears here for the first time. The paper concludes with a discussion of history and literature. NEWLINENEWLINENEWLINEThe author has provided a valuable service in writing this paper. It is already helping experts understand definitions of others and even clarify their own. It will also help students to enter the topic.