This is a hasty reply to David Yetter's question. The original Dold-Kan theorem is that the category of simplicial abelian groups is equivalent to the category of chain complexes of abelian groups. Dominique Bourn has studied this quite deeply. One of his results (all published in JPAA & Cahiers) is that the category of length n chain complexes of abelian groups is equivalent to the cat of abelian groups in n-Cat (= n-categories in Ab). Thus, looking at homs, we can relate differential graded categories (DG-categories are categories with homs in chain complexes of abelian groups) and (n+1)-categories. However, there is a need to look at the tensor products on these base monoidal categories. This is why David was led to his next question which Bourn has solved for all n in the additive case. In the non-additive situation, we need the tensor product of John Gray on 2-Cat, not the cartesian product. Categories enriched in this monoidal category are almost 3-categories except for a wobbly middle-4-interchange. Groupoid- like such are Joyal-Tierney homotopy 3-types. Higher dimensional cases are at various stages of completion in the non- additive case: Bourn has done the additive case: I have preprints of his papers "Un theoreme de Dold-Kan pour les groupes abeliens n-categoriques", "The tower on n-groupoids and the long cohomology sequence" JPAA 62 (1989) 137-183 "Produits tensoriels coherents de complexes de chaine" Bull soc Math de Belgique 41 (2) (1989) 219-248 "Another denormalization theorem for abelian chain complexes" JPAA Regards to all: sorry I won't get to Sussex this year. Ross Street ++++++++++++++++++++++++++++++++