I've been waiting for one of the people more expert in the subject than I to answer Vaughn's question. Since no one has, I would like to make some comments on it. The remark that he supposed it couldn't be a cartesian product misses the point. A cartesian product is unique (up to unique isomorphism) and if the cartesian product doesn't have the right properties, then a product that does certainly isn't the cartesian product. This product wouldn't be in the category of n-categories, since you want the product to be a 2n-category. You presumably want the category of omega-categories or at the most those which are actually n-categories for some n. Third, the analogy with the story breaks down. To be analogous, you should be looking at the tensor product of two stories to have 6 parts, not nine. You would have to identify <begin, end>, <middle, middle> and <end, begin> and there would seem to be little reason to expect this to make sense. In the additive case, there is some reason for thinking this might make sense, but not in general. What seems likelier is that the tensor product of two categories is a two dimensional category. This is not at all like a 2-category; rather it is as though each entity was at once an i-cell for one omega-category structure and a j-cell for another one. Of course, you would then have to allow three dimensional, four dimensional, ... . My intuition on this comes from simplicial sets, where there is a tensor product of two simplicial sets to get a double simplicial set. Aside from the diagonal simplicial set (which is the cartesian product) you can also, in the additive case, take the associated chain complex and then construct a simplicial set from that, which is a kind of cartesian product. This is homotopic to the diagonal, but I imagine it is still different from it. In the general case, this makes no sense. Michael Barr