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. 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...

The theory of directed complexes is extended from free !-categories to arbitrary !- categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !-category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !-categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...

The pasting theorem showed that pasting schemes are useful in studying free !-categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !-categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !-category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !-categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...

§1 Types, shapes and occurrences p. 3 §2 Factorization and geometry p. 13 §3 Cuts in partial orders, and planar arrangements p. 32

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Multiple categories: the equivalence of a globular and a cubical approach

We show the equivalence of two kinds of strict multiple category, namely the wellknown globular o-categories, and the cubical o-categories with connections. # 2002