Abstract

. In the literature there are several kinds of concrete and abstract cell complexes representing composition in n-categories, !-categories or 1-categories, and the slightly more general partial !-categories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `!-complexes' which consists of all complexes representing partial !-categories. We show that !-complexes can be given geometric structures and that in most important examples they become well-behaved CW complexes; we characterise !-complexes by conditions on their cells; we show that a product of !-complexes is again an !-complex; and we describe some products in detail. 1. Introduction In this paper we consider pasting diagrams representing compositions in multiple categories. To be specific, the multiple categories concerned are n-categories and their infinite-dimensional analogues, which are called !-categories or 1-cat...

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