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Implementing finite structures in Mathematica via a skeletal topos of finite sets

Susan B. Niefield
Organization: Union College
Department: Mathematics
URL: http://www.math.union.edu/people/faculty/niefiels.html
Journal / Anthology

Journal of Symbolic Computation
Year: 2003
Volume: 35
Page range: 137-151

Abstract To implement finite structures in a symbolic computation program such as Mathematica, we consider a skeletal topos N which is equiva- lent to the category Set f of finite sets. Objects of N are nonnegative integers, and morphisms f : n \Gamma ! m are lists (f1; : : : ; fn) of integers such that 1 ^ f i^ m, for all i. A full and faithful functor from N to Set f is obtained by identifying n with the set [n] = f1; : : : ; ng and identifying (f1; : : : ; fn) with the function i 7! fi. A topos structure on N (appropriate for Mathematica) is obtained by transporting the topos structure of Setf along a suitable pseudo-inverse C of the functor from N to Set f described above. The code for the Mathematica implementation included below is also available as a Mathematica Notebook [6].

*Mathematics > Foundations of Mathematics > Set Theory