What is the depressing thought Vaughan? Thank you for the nice comments about the "oriented simplexes" paper. You have clearly read it in depth. When you were in Sydney a couple of years ago, you gave a talk on concurrent programming (in a room we attended many times as undergraduates together). After it I tried to explain a little about parity comp lexes. I had (and still do have) the feeling that they provide a convenient notion of rewrite system in the parallel programming situation. The terms to be rewritten are not a priori ordered, the possible orders only come out when the rewrite is running. When I mentioned the connection with rewrites you suspected I was only looking at Turing machines, or the equivalent. So I am extremely happy that you see some connection now with some higher tensor product. I have sat on the parity complexes paper on the recommendation of Sammy Eilenberg. There is room for improvement, some of which I, and Mike Johnson , have done. However, there is no doubt that the basic structure is correct, the only question involves the axioms. The basic structure is a graded set C with, for each element x of dimension n, two finite subsets x+ and x- of elements of dimension n-1. The axioms I give in the Report are strong enough to ensure sufficient loop-freeness to allow the basic simple construction of an omega category much as in "oriented simplexes" to work. Also the axioms are closed under Product (as defined there). This gives the example of cubes for free, and the example needed for higher descent obtained as the product of globs with simplexes. I drew a picture of the product G?2?xGreekD?2? of the free 2-cell and the triang le on 20 Jan 1988. I thought I sent you one. Anyone who wants it can ask; I don' t think I'll tex it into this message. The point is that the basic structure is simple, as described above, and the product is simple too; if X, A are parity complexes, the product is XxA graded by dim(x,a) = dimx + dima, and with (x,a) = xsuperplus cross {a} union {x} cross asuperplusorminus(depending on dimx). There is no need to read 15 pages to get to this!!! Then you can draw diagrams and examples of your own choosing: ab to ba, or vice versa. For some reason, Iain and I built up our cubes from the basic interval (+) arrow pointing right (-). I believe this differs from the direction in Iain's "oriented cubes" Report. The tensor product you seem to want is also related to Gray's tensor product of 2-categories. You see, you can approach that tensor product by the desire that the arrow category tensor itself should be a square with a 2-cell in it. The trouble with restricting to 2-categories is that to make (2-category) tensor (2-category) come out a 2-category, and not a 4-category, we must kill off higher order cells. So enter word problems in braid groups etc. Life is much simpler if we just allow our dimensions to escalate. Back to the question of my parity complexes axioms. They are NOT definitive. (The structure is, as I said before.) I designed them for the two purposes mentioned above: to get the free 2-category by the "obvious" simple requirement of well-formedness; and, to get closure under product. I conceitedly hoped that those two goals might force the definitive concept. John Power burst that bubble by showing an example of everyone's idea of a pasting diagram which did not satisfy my (fourth, I believe) axiom. There is every reason to believe however that Mike Johnson's "Pasting schemes" are the definitive answer. There seems to be some new word on that seen: a (oops misspelt "scene" on last line) simplification of Johnson's axioms by the Moscow School (announced at the Cambridge Meeting recently). But don't worry about all that. Products of globs satisfy everybody. Best regards, Ross.