This is something of a reply to Vaughan Pratt's letter of 13 Nov on homotopies as hom-objects. This is fairly well used in the case of groupoids. It is mentioned in my survey `From groups to groupoids' Bull London Math. Soc. 19, 113-134, 1987. In the case of multiple categories, the situation is more complicated. Ehresmann and Ehresmann studied this in papers in the Cahiers. Brown and Higgins studied the cubical version of mutiple categories and the internal tensor product and hom in JPAA 47, 1-33, 1987. The main interest in that paper was, however, carrying this structure over, in the groupoid case, to the equivalent cate- gory of crossed complexes. The aim in any case is that the internal hom carries information on homotopies and higher homotopies. Infinity categories were defined by Brown and Higgins in Cah. Top. Geom. Diff. Cat. 22, 371-386, 1981. There, infinity groupoids were shown equivalent to crossed complexes. Richard Steiner (Glasgow) and F.Al-Agl (Saudi Arabia, ex Bangor) have recently studied the relations between infinity categories and various forms of cubical or simplicial infinity or multiple category. The aim is to obtain equivalences of categories, so that there is transfer of information from one category to the other. This has been important in the applications of the groupoid case to homotopy theory. This area is summarised in R.Brown, `Some problems in non-Abelian homotopical and homological algebra', Springer LNM 1418, 1990, 105-129. The cubical case is an easy one for defining homotopies and higher homotopies, and the adjoint tensor product. There is work to be done on say the simplicial case. Crossed complexes are equivalent to simplicial T-complexes, (Ashley, Diss. Math. 265, 1988), but no-one has produced the internal hom and tensor product in that category. Ronnie Brown