ar X iv : q - a lg / 9 70 20 14 v 1 1 0 Fe b 19 97 Higher - Dimensional Algebra III : n - Categories and the Algebra of Opetopes

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 I0+ = I, I(i+1)+ = (Ii+)+, 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 ωcategories. In a similar manner, 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 ncoherent O-algebra objects may be defined in any (n + 1)-coherent O-algebra.