NERVES OF MULTIPLE CATEGORIES FAHD ALI AL-AGL and RICHARD STEINER [Received 15 February 1991—Revised 28 November 1991] 1. Introduction and statement of results The multiple categories studied in this paper are ^-categories, defined as follows. DEFINITION 1.1. An ^-category is a set X together with unary operations do, do, dx, dx,... and not-everywhere-defined binary operations x 0 , Xj,... such that: (1.1.1) for n = 0, 1,..., there is a category C n such that the morphisms of C n form the set X, the left and right identities of an element x of X are d~x and d*x, and composition is x n ; (1.1.2) for m # n the operations d~, d^ and x m commute with d~, d* and ><„; (1.1.3) for m<n, each identity of C m is also an identity in C n ; (1.1.4) each element of X is an identity in some C n . A more explicit form of this definition is given in Definition 2.1. The concept was originated by Brown and Higgins in [5], in connection with homotopy theory. Their definition did not include (1.1.4), but it is commonly included in practice. The concept without (1.1.4) has also been studied by Street [16] under the name of co-category, for use in connection with cohomology theory. For a fixed natural number n, an <»-category in which each element is an identity for x n is called an n-category; there has been a lot of work on 2-categories. In this paper we study geometric models for oo-categories. We shall show that oo-categories are equivalent to cubical sets with additional structure or to simplicial sets with additional structure. The cubical case was suggested by an analogous result of Brown and Higgins for <»-groupoids [5]. The simplicial case answers a question of Street [16, 5], though rather inexplicitly. As an application, we show how the cartesian product of cubes yields a biclosed monoidal structure on the category of oo-categories; this is related to work of Gray [8,1,4] and Ehresmann and Ehresmann [7, D2 Complement 2]. Pratt [14] has suggested that oo-categories might serve as a model for concurrency in computing, and tensor products would be important in this theory. They may also help in understanding such topics as homotopies and lax functors: we associate an oo-category G(I N ) to the yV-dimensional cube I N (see Theorem 1.7 below), and tensor products with the G(I N ) could serve as domains of higher homotopies. The general approach is as follows. Throughout the paper, let W be a Hausdorff topological space and let w 0 , w u ... : W^U be continuous functions. Let x be a subset of W and let n = 0, 1, ... . An n-fibre of x is a non-empty subset 1991 Mathematics Subject Classification: 18D05. Proc. London Math. Soc. (3) 66 (1993) 92-128.