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 <= 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.

Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case.

Poly-categories form a rather natural generalization of multi-categories. Besides the domains also the codomains of morphisms are allowed to be strings of objects. Multi-categories are known to have an elegant global characterization as monads in a suitable bicategory of special spans with free monoid as domains. To describe poly-categories in similar terms, we investigate distributive laws in the sense of Beck between cartesian monads as tools for constructing new bicategories of modi ed spans. Three very simple such laws produce a bicategory in which the monads are precisely the planar poly-categories (where composition only is de ned if the corresponding circuit diagram is planar). General poly-categories, which only satisfy a local planarity condition, require a slightly more complicated construction.