In my previous talk, Max reminded me that there are precisely two monoidal biclosed structures on Cat: the cartesian product, and what Ross calls the "funny" tensor product. Enriching over these gives 2-categories and sesqui- categories respectively. One can envisage a generalization of this to higher dimensions, with more monoidal structures, e.g. Gray's tensor product of 2-categories, and with more ways of enriching, e.g. some sort of "weak" enrichment, e.g. one which should give bicategories and tricategories when enriched. This led to the following tentative heuristic hypothetical diagram of higher-dimensional categorical structures, where the lines indicate enriching over possible tensor products: Sets | -- Cat - -- / | \ \ \ - Sesqui-Cat -- 2-Cat Bicat / / | / | / | \ . - 3-Cat Gray-Cat Tricat / / / | \ / | \ / | \ . . . . / / | \ proto-Teisi/NIC . . . ω-Cat . . . ω-Teisi . . . . weak-ω-Cat The following small parts of this diagram have hitherto be considered: up to sesqui-categories, the whole line to ω-categories, and the Gray-tensor on it (see Chapter 3 of my PhD thesis), up to 4D teisi (with the results in this series of talks), and the named structures in the right hand line of the diagram.
I proved that the tensor product of Gray-categories defined a couple of talks ago is associative, via a presentation for the triple tensor. See sections 6 and 7 of the paper "A tensor product for Gray-categories".
Re: Max' question whether proto-teisi/no-interchange categories can be defined in terms of a higher-dimensional version of the "funny" tensor product on Cat: define a no-interchange category as globular set C together with "vertical" compositions, where the dimension of the intersection of the elements to be composed is exactly one less than the dimension of the elements, and "whiskerings", where one of the elements is of higher dimension, subject to the usual axioms. Define a tensor product of NIC's by having as generators pairs (c,d) with one of these 0-dimensional, and relations coming from composition in each variable. Multiple tensor has generators (c_1, ..., c_k) with at most one of these not 0-dimensional. Note that the internal hom for this tensor product has as elements maps of degree k without any naturality condition. Theorem: a NIC-enriched category is a no-interchange category.
Re: Giulio's question on whether double categories can be fitted in the current set-up: Given two categories C and D there is an obvious double category which is C horizontally, D vertically, and has squares pairs (f,g) with obvious faces and compositions. The 2-category associated to this double category is precisely the Gray-tensor of C and D (viewed as 2-categories). Given two 2-categories C and D there is, similarly, an obvious quadruple category. But there is only a 4- (and by reflection a 3-)category associated to this, not a Gray-category, because multiple categories have a strict interchange law.