Higher dimensional groupoids One reason to see how higher dimensional categories exist and are useful is to look at the groupoid case. Higher dimensional groupoids form a useful higher dimensional version of groups and the fundamental group. The old result that "a set with two compatible group structures is just an abelian group" (basically, 1932, with Cech's description of higher homotopy groups) was for long regarded as an obstruction to such a programme. In fact, sets with 2 compatible groupoid structures model 2-dimensional homotopy theory, and so on for n compatible groupoid structures. This makes them of course very complicated, even in dimension 2. This fact is interesting in itself. A clear problem was to define a higher homotopy groupoid. One solution was by the writer and PJ Higgins, as may be found in joint work in (Proc London Math Soc 1978, for dimension 2, JPAA 1981 for all higher dimensions). This gives what we called the fundamental omega-groupoid of a filtered space. Later, Loday found a more complex and more general model, the fundamental cat^n-group of an n-cube of spaces. The existence of these functors is non-trivial. My purpose in considering higher dimensional groupoids since about 1966 was the possibility of higher dimensional van Kampen Theorems, modelled on the groupoid version. The references are the above papers with Higgins, and also with J-L Loday (Topology, 1987, Proc London Math Soc 1987), with his more complicated algebraic model. These theorems and models now allow for specific computations of some homotopy invariants and even some homotopy types (my paper in the Adams Memorial Colloquium). The intuition is that one needs algebraic structures which can model the geometric notion of subdivision, and which have structure in a range of dimensions, to carry the information about how bits of a space fit together. Multiple groupoids (or categories) seem well placed for that.The intuition is related to old ideas in topology of "What is a cycle?". A cycle should be some kind of composition of the little pieces. How should one accomplish this, algebraically? One has to move away from a linear notation and an always defined composition. Taking the free abelian group on the little bits seems like a cop-out (but of course, it has its uses!). One aspect of the theory is the equivalence of various views of a given structure. This is referred to by Ross Street in his communication, with regard to infinity-categories. So we have a set of equivalences between crossed complexes, omega-groupoids (RB-PJH, JPAA 1981), infinity-categories (CTGDC 1981), cubical T-complexes (CTGDC 1981), simplicial T-complexes (Ashley's thesis of 1978, published in Diss Math 155, 1988), polyhedral T-complexes (Jones, 1984, published as Diss Math 156, 1988). Here T-stands for "thin". These canonical structures are a generalisation of identities which appear only from dimension 2. The idea appears in old axioms for groups; with Keith Dakin in Bangor in 1975; and independently with Roberts in Australia at the same time, with the name "hollow". Thin elements play a crucial role in computations in and manipulations with these objects. Intuitively, thin elements have commutative boundary, and commutative boundaries have thin fillers. Thin elements also define compositions: all faces but one of a thin element "compose" to give the remaining face. For more on the philosophy of this, see Jones' thesis, which deals with compositions of general faces. Even more complex is the equivalence between cat^n-groups and crossed n-cubes of groups (Ellis-Steiner, JPAA, 1987). These equivalences allow a translation from a linear notation to a "higher dimensional" notation. Verity's equivalence in the category case mentioned by Ross Street, is in the same spirit as, and generalises, the equivalence between simplicial T-complexes and infinity-groupoids (go through crossed complexes, combining RB-PJH with Ashley). These things work. I just realised that one can do many sums with results of the first paper of RB-PJH, which for example allows the computation not only of \pi_2(BH \cup CBG) when G is a subgroup of H, for specific G,H, but also of the first k-invariant, i.e. the 2-type, of this space. These ideas are all modelled on groupoids, as is natural for the homotopy applications and the Generalised Van Kampen Theorems. A categorical version has not been done in full generality (polyhedrally). One does not expect the same type of application. There are relations with cohomology and knot theory which need more investigation. A lot of these ideas were discused in the visit of Ross and Dominic Verity to Bangor in May/June, 1993. Ronnie Brown ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++