Z tensor product for (Gray-Cat)_tensor-categories

Sjoerd Crans – 25 March 1998

First, Y tensor product? For the stricter counterpart of weak n-categories, which will be useful for proving things, via the ``Kelly-approach'', and for computation. Also for the classification of homotopy n-types; even if iso-(Gray-Cat_tensor)-categories don't classify 4-types, what do they classify then? The basic principles for higher dimensional Gray-categorical structures are: horizontal composition is dimension raising, governed by the topological product of globes, axioms will be functoriality and associativity, labeled by ``elementary'' pasting schemes (or, equivalently, trees), heuristically axiom equates ``most economical decomposition'' with ``most uneconomical decomposition''. In practice, a 5-dimensional tas consists of a graded set (C_i)_i together with operations n-source, n-target: C_i -> C_n, n-composition: C_q x_n C_p -> C_{p+q+n-1}, and identity C_i -> C_{i+1}, such that source, target, vertical composition (i.e., where one of p, q is n+1, so this includes whiskering) and identity behave as in an omega-category, and such that horizontal composition behaves as follows: for c' and c n-composable, with source (c)=a and target (c')=a'' say, the faces of c' comp_n c are given as a certain composites in C (a, a''). For example, for p=q=3 and n=0 the diagram is as on page 13 of [tpgc] (but with different labeling), and involves other horizontal composites, and their inverses. functoriality: for example, -- / \ --- .----. . \ / --- -- gives rise to d comp_0 (c' comp_1 c) = (d comp_0 c') whiskerandcomp_1 (d comp_0 c). This becomes very complicated in higher dimensions (I gave a 5-dimensional example on a slide), involving more horizontal composites and their inverses, and involving lower-dimensional axioms, to make both sides close up. Sometimes it only involves lower-dimensional axioms, in which case it holds trivially. associativity: for example, --- --- --- . . . . --- --- --- gives rise to the familiar Yang-Baxter diagram. interchange: does not follow automatically, and needs to be included. also need functorio-associativity, arising from -- --- / \ --- . .----. . , --- \ / --- -- cf. the diagram on page 46 of [tpgc], and functorio-functoriality. Problem 1: how to determine where horizontal composites occur in the faces of a composite and in the axioms, and which direction they have. Problem 2: how to prove that all axioms close up. Problem 3: how to recognize which axioms follow from lower-dimensional ones. Non-problem 4: composition is a functor C(a,a') ``tensor'' C(a',a'') -> C(a,a''), but this tensor product (of (Gray-Cat_tensor)-categories) is not associative. So, one could say this is not A, i.e., Z, tensor product. Still, there should be a suitable, post-modern, way to enrich with respect to it resulting in 5-dimensional teisi.