A Forbidden-Suborder Characterization of Binarily-ComposableDiagrams in Double Categories
Robert J. MacG. Dawson · Theory and applications of categories · 1995
. Tilings of rectangles with rectangles, and tileorders (the associated double order structures) are useful as "templates" for composition in double categories. In this context, it is particularly relevant to ask which tilings may be joined together, two rectangles at a time, to form one large rectangle. We characterize such tilings via forbidden suborders, in a manner analogous to Kuratowski's characterization of planar graphs. 1. Introduction A double category, D, is a category object in Cat. (The concept of a double category was first introduced by Ehresmann [3] in 1963.) As such, it can be thought of as consisting of a collection D of objects, a collection D h of horizontal morphisms, a collection D v of vertical morphisms, and a collection D2 of double morphisms ("cells"). hD; D h i is a category; so are hD; D v i, hD h ; D2 i, and hD v ; D2 i. Thus, the elements of D2 have two compositions: the "horizontal" composition which D has as an object of Cat and the "vertical" composit...