DINATURAL NUMBERS

The notion of strong Barr dinatural transformation is introduced which , when taken between Hom functors, gives a notion of natural number sp ecifically adapted to the category under consideration. We call these dinatural numbers and we study their arithmetic which depends in a nic e way on the structure of the category. We also consider families of d inatural numbers, which leads to a new universal property for natural numbers object as classifying object for dinatural numbers. When there is a natural numbers object, its arithmetic defined by recursion corr esponds to the arithmetic of dinatural numbers. Examples are given wit h a particular emphasis on the category of finite sets. (C) 1998 Elsev ier Science B.V. All rights reserved.