I've been working for several months on an approach to higher dimensional composition and the accompanying higher dim category theory (functors, natural transformations, limits, etc...). (It may be related to Vogt's work, cited in Barr's reply, but I'm unfamiliar with that work.) I'd be happy to send a "work-in-progress" report (in Latex format) giving details on what I've discovered thus far. The "dimension 1" cases are ordinary categories. In the dimension n case: objects are (n-1)-simplexes, maps are n-simplexes and composites of maps are (n+1)-simplexes of a simplicial set C. As Barr described in his response, the partial binary operation which defines composition in an ordinary category, associates a map (= 1-simplex) to a pair of maps whose source and target match appropriately. In the n-dim case there is a partial (n+1)-ary operation which, given n+1 n-simplexes whose faces match appropriately, produces their "composite", another n-simplex whose faces match the given ones according to the simplicial identities. One has to specify *which* of the faces of the n+1 simplex is the composite of the others. It can be the face opposite vertex i for any i = 0,..., n+1. The most efficient way to specify all this is by saying that the standard function from C_{n+1} ( = the n+1 simplexes of C) to the set of open i-boxes of C of dimension n+1 is an isomorphism. Examples (though special cases) of such structures are already known: n-dimensional hypergroupoids. These include Eilenberg-MacLane spaces and also arise from the singular complex of any topological space. Rreferences to hypergroupoids include my 1982 paper ("Realization of Cohomology Classes in Arbitrary Exact Categories", Jour. of Pure and Appl. Alg. 25 (1982) 33-105) and papers by Jack Duskin. There seems to be a reasonable "category theory" for this kind of higher-dimensional composition. For example, functors C --> C' and natural transformations of such functors occur as simplexes in the function complex C'^C (of dimensions n-1 and n respectively). If C and C' are higher dim categories in the sense I described above, then so is C'^C. Paul Glenn Dept. of Mathematics Catholic University of America Washington, DC 20064 Internet: glenn@cua.edu Phone: 202 319 5221 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++