Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes

Abstract We give a definition of weak n -categories based on the theory of operads. We work with operads having an arbitrary set S of types, or “ S -operads,” and given such an operad O , we denote its set of operations by elt( O ). Then for any S -operad O there is an elt( O )-operad O + whose algebras are S -operads over O . Letting I be the initial operad with a one-element set of types, and defining I 0+ = I , I ( i +1)+ =( I i + ) + , we call the operations of I ( n −1)+ the “ n -dimensional opetopes.” Opetopes form a category, and presheaves on this category are called “opetopic sets.” A weak n -category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weak ω -categories. Similarly, starting from an arbitrary operad O instead of I , we define “ n -coherent O -algebras,” which are n times categorified analogs of algebras of O . Examples include “monoidal n -categories,” “stable n -categories,” “virtual n -functors” and “representable n -prestacks.” We also describe how n -coherent O -algebra objects may be defined in any ( n +1)-coherent O -algebra.