In the literature there are several kinds of concrete and abstract cell complexes representing composition in n-categories, \omega-categories or \infty-categories, and the slightly more general partial \omega-categories. Some examples are parity c omplexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `\omega-complexes' which consists of all complexes representing partial \omega-categories. We show that \omega-complexes can be given geometric structures and that in most important examples they become well-behaved CW complexes; we characterise \omega-complexes by conditions on their cells; we show that a product of \omega-complexes is again an \omega-complex; and we describe some products in detail.
Keywords: pasting diagram, n-category, .omega-category, infinite-category, partial omega-category, parity complex, omega-complex, directed complex.
1991 MSC: 18D05.
Theory and Applications of Categories, Vol. 4, 1998, No. 1, pp 1-36.