Sjoerd Crans – 25 June 1997
The basis for the mix was a look at John Power's 4-dimensional counter example, a pasting which is neither a pasting scheme, a 4-pasting scheme, a parity complex nor a directed complex. I described its geometrical structure, which is based on an Escher staircase, and its combinatorial structure, which shows that the possibility of changing the order of composition of the 3-dimensional intersection of the pasting is the main culprit.
Question: is this counter example a Buckland scheme? I.e., is the paste a legitimate one?
First solution for the counter example: pasting presentations, as in Chapter 2 of my PhD thesis "On combinatorial models for higher-dimensional homotopies". The possibility of inserting identities makes that the pasting can be done without introducing any loops.
Second solution: 4-dimensional teisi. 3-dimensional teisi are Gray-categories, which differ from 3-categories in that horizontal composition of 2-cells is not definable in terms of vertical composition, but that there is a comparison-3-cell between the (two) possibilities. Using similar comparison-cells in dimension 4, the loop in the counter example disappears.
It would be nice if there were a way of codifying this order-of-composition information in the notion of scheme. This leads to the (vague) idea of pswr, pronounced "psoor", which are supposed to be pasting schemes with extra order relations between the cells telling in which order they are to be (vertically) composed, with the absence of such relation indicating the higher-dimensional comparison cell.