Representing numeric gadgets over weak number objects

Harold Simmons (Manchester)
24 October 2001

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Dedekind's characterization of the natural numbers can be given a categorical setting. In a suitable category (cartesian closed) the natural number object is the essentially unique object that supports enough recursion and induction (in the sense that it carries a unique mediating arrow to any other potential numeral structure). By dropping the uniqueness requirement we obtain a weak number object. This supports recursion but not induction, and a category can have many different such objects.

Given such an object any numeric gadget constructed by recursion can be modelled in the category. In this way we can get exotic versions of arithmetic. I will explain how such objects are used and show the connection with some non-categorical ideas.

University of Leeds, Department of Pure Mathematics, Logic Seminar 2001