Categories beyond globes and cubes

Abstract:
We study the functorial dependence of the special category theory upon the shapes of cells to be composed. In particular, we consider the categories, whose dimensions form ordered set. Examples and applications of categories of different shapes are discussed:
    a) cyclic orders, adjunctions and closed structures;
    b) cubes and A_¥-categories;
    c) Markl's special PROPs as a new categorical shape.