A Cellular Nerve for Higher Order Categories

Introduction The following text arose from the desire to establish a rm relationship between higher order categories and topological spaces. Our approach combines the algebraic features of Michael Batanin's !-operads 1] with the geometric features of Andr e Joyal's cellular sets 15] and tries to mimick as far as possible the classical construction of the simplicial nerve of a category. Higher order categories have attracted much attention in the last decade, due to their appearance in several mathematical areas. The ultimate goal is perhaps a faithful algebraic description of homotopy systems 13]. Since a homotopy between homotopies has the shape of a disk, the next higher homotopy the shape of a ball, and so on, we chose \ball compexes", i.e. globular sets 26], as the primitive combinatorial objects. The globular structure is precisely what underlies an !-category 26], once its multiplicative structure is forgotten. Conversely, an !-category can be deened as a globular monoid, in the same way as a small category can be deened as a graphical monoid. An innnite-dimensional endomorphism-cell freely generates an !-category whose cells can be labelled by nite planary level trees. This is the basis of Batanin's concept of a weak !-category. The precise deenition relies on a new notion of operad, acting on globular sets, and called !-operad. Because of the importance of !-operads, our exposition begins with a proof that each !-operad A embeds in a homogeneous globular theory A. The existence of such an embedding goes back to Michael Boardman and Rainer Vogt's deenition 5] of operads in terms of algebraic theories. A weak !-category is then an algebra over a contractible !-operad A or, equivalently, a A-model, i.e. a presheaf X : op A ! Sets whose restriction to the underlying globular site is a sheaf. The terminal homogeneous theory for globular sets acts on !-categories and, remarkably, coincides with Joyal's cell category 15]. There is even a level ltration of by subcategories (n) acting on n-categories. For n = 1, we get the simplex category = (1) which acts on categories and is thus the terminal homogeneous theory for graphs. As immediate consequence, we obtain a cellular nerve for !-categories, which extends the well known simplicial nerve for categories. Cellular sets, i.e. 1 presheaves on , play an important double r^ ole (completely analogous to sim-plicial sets) : on one side, the-models are precisely the nerves of !-categories; on the …