In the groupoid case, there are various possibilities for `higher dimensional objects', namely crossed complexes, infinity-groupoids, omega-groupoids, simplicial T-complexes, poly-T-complexes (see the references in my article on `Some problems in non-abelian homological and homotopical algebra' SLNM 1418, ed M Mimura). Crossed complexes are useful because of their close relations to classical tools (chain complexes, universal covering spaces, relative homotopy groups). Omega-groupoids (=cubical infinity-groupoids with connections, and so thin structures) are useful because of the clear compositions, monoidal closed structures, and context for proving the generalised Van Kampen Theorem. Simplicial T-complexes are useful because of the relation with classical simplicial techniques, and because of a lingering suspicion in some quarters that it is old fashioned to use cubical sets. The aim of the poly-theory was to to give a context for group theory methods such as Van Kampen diagrams (see D.Johnson, Presentations of groups, LMS Lecture notes). In any case, what is wrong with pentagons, or rhombic dodecahedra? How do Stasheff polyhedra fit into a general theory similar to simplicial sets? What algebraic structures do Stasheff T-complexes correspond to? What is not so clear is what are the valuable features of infinity-groupoids. It is somehow interesting to know they are there, but what does one do with them when one has got them? I wish I knew. For example, translating the monoidal closed structure on omega-groupoids to the infinity-groupoid case, with explicit formulae, seems a formidable task. (It is not so good for the simplicial case, either!) There are also problems. For example, what is the crossed complex corresponding to the free infinity-groupoid on one generator of dimension n? It should be the fundamental crossed complex of the n-cell with its globular (hemi-spherical ) subdivision. But how does one prove this? The corresponding results for the cubical and simplicial case use basic facts on the homotopy theory of geometric realisations for the (current) proofs. A paper of Spencer and Wong (CTGD 24 (1983) 164-192) shows the advantage of working with double categories with thin structure rather than with 2-categories. Part of the aim of the thesis of F. Al-Agl was to get similar methods in the n-dimensional case. Intuitively, one hopes to replace pasting arguments by calculations with thin elements, since these obey the transparent rule: any composition of thin elements is thin. One can also make inductive arguments about how degenerate is a thin element (see Proposition 4.4 of Brown-Higgins JPAA 22 (1981) 11-41: the whole theory of omega-groupoids is designed to make this Proposition work, since the immediately following lemma gives the crucial result that certain choices previously made do not affect the final composition, and this is proved by showing that an (n+1)- dimensional cube is degenerate in direction n+1, so that its two opposite faces in this direction are the same). Trying to do this last proof of the Van Kampen Theorem in infinity-groupoids is a nightmare of intricately coiling tubes (see Whitney's tube systems?), so that the work with Loday on cat-n-groups adopted an entirely different approach, using sophisticated simplicial techniques, which in some mysterious way accommodate all these local problems. Clearly n-categories arise in nature. But it would seem useful to assess the advantages and disadvantages of the various possible categories in which to work before committing oneself to setting up a homotopy theory in one setting rather than another. The advantage of having these (highly non-trivial) equivalences of categories is that one can swap round at will, not noticing the technical nature of the machine which allows one to do this. But as Vaughan Pratt emphasised, there is still the problem of proving that infinity-categories are equivalent to omega-categories (=cubical infinity-categories with connections), and this is not yet solved by the work of Al-Agl/Steiner, although that does give a beautiful cubical setting equivalent to infinity-categories. I was very interested to see the high priority given by Vaughan to this question. Ross Street in an extra informal lecture at Montreal explained the background in constructing `cohomology with coefficients in an n-category' for his work on nerves of infinity-categories (following up suggestions of John Roberts). I have tended to concentrate on the corresponding possibilities for the groupoid case, but work of Pachkoria explained at Montreal shows the geometric possibilities of cohomology with coefficients in a commutative monoid. Recent work of Larry Breen gives Schreier systems with coefficients in a crossed square, while Bullejos and Cegarra have considered coefficients in a reduced `crossed module of length 2' (a la Conduche). Brown-Higgins deal in a recent preprint (`The classifying space of a crossed complex', MPCamb Phil Soc, in press) with coefficients in a crossed complex. It all looks very promising. Ronnie Brown ===================================