The nerve of a groupoid (not just a group) is also a Kan complex in the strong sense of a simplicial T-complex (Keith Dakin's definition, 1975). A simplicial T-complex (K,T) has sets T_n in K_n of elements called thin with the property that: 1) any horn has a unique thin filler 2) degenerate implies thin 3) if all faces but one of a thin element are thin, so is the last face. This is of rank less than or equal to n if all simplices in dimensions greater than n are thin. Simplicial T-complexes of rank 1 are equivalent to groupoids (Dakin). Simplicial T-complexes in general are equivalent to crossed complexes (Ashley, see Diss Math 265 (1988) Simplicial T-complexes and crossed complexes; Nan Tie, JPAA 56 (1989) 195-209), to infinity-groupoids, and to other things. The notion of cubical T-com plex is crucial in the proof by Brown-Higgins of an n-dimensional Van Kampen Theorem (JPAA 22 (1981) 11-41), because of the technical point that it easily enables the handling of multiple compositions of homotopy addition lemmas (the boundary of a simplex or cube is the "sum" of its faces) without writing down formulae. For polyhedral versions, see D W Jones, Diss Math 266 (1988) A general theory of polyhedral sets and the corresponding T-complexes.) Part of the point is that the singular complex SX of a space X is a Kan complex but the fillers come from the models, so ought, morally, to be canonical and to satisfy relations. But these relations are up to homotopy, it seems. So it is difficult to be more precise. The T-complex condition is very strong, but is nice in that it is easy to see how to weaken it. These weakenings need lots more investigation. A further point is that a filler of a horn of a triangle determines a product, as if it were a `computation'. Analogous notions for categories are studied by Ross Street and by Dominic Verity. Ronnie Brown Prof R. Brown Tel: (direct) +44 248 382474 School of Mathematics (office) +44 248 382475 Dean St Fax: +44 248 355881 University of Wales email: mas010@bangor.ac.uk Bangor wwweb for maths: http: //www.bangor.ac.uk/ma Gwynedd LL57 1UT UK