The continuum as a final coalgebra

Abstract

We define the continuum up to order isomorphism, and hence up to homeomorphism via the order topology, in terms of the final coalgebra of either the functor N×X, product with the set of natural numbers, or the functor 1+N×X. This makes an attractive analogy with the definition of N itself as the initial algebra of the functor 1+X, disjoint union with a singleton. We similarly specify Baire space and Cantor space in terms of these final coalgebras. We identify two variants of this approach, a coinductive definition based on final coalgebraic structure in the category of sets, and a direct definition as a final coalgebra in the category of posets. We conclude with some paradoxical discrepancies between continuity and constructiveness in this setting.