The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n is less than or equal to 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Keywords: n-category, operad, higher-dimensional category
2000 MSC: 18D05, 18D50, 18F99, 18A99
Theory and Applications of Categories,
Vol. 12, 2004,
No. 3, pp 73-194.