Monoidal categories with natural numbers object

The notion of a natural numbers object in a monoidal category is defined and it is shown that the theory of primitive recursive functions can be developed. This is done by considering the category of cocommutative comonoids which is cartesian, and where the theory of natural numbers objects is well developed. A number of examples illustrate the usefulness of the concept.This work was partially carried out while both authors were guests of McGill University and while the second author was a guest of Dalhousie University. Both authors acknowledge support from the Natural Sciences and Engineering Research Council of Canada and the Québec Department of Education.