Re: Constitutive Structures

[Vaughan Pratt <pratt@cs.stanford.edu>]

One answer to this question is that the continuum is the final
F-coalgebra in a suitable category for a suitable F.  The first result
along those lines was Pavlovic & P, "The continuum as a final
coalgebra", TCS 280(1-2):105-122, May 2002, originally presented at
CMCS'99 in Amsterdam.  It made explicit the double coinduction implicit
in the various continued-fraction representations of the reals.  The
category was Posets and only the topological and order structure was
represented.  This was subsequently extended in papers by Peter Freyd
and by Tom Leinster to express as well the algebraic structure, and also
to reduce the double coinduction to a single coinduction in exchange for
giving up uniqueness of representation of reals (the continued fractions
are in bijection with the nonnegative reals).

Vaughan Pratt

On 4/7/2011 5:50 AM, Ellis D. Cooper wrote:
> What might be the proper categorical framework to discuss, for
> example, the fact that the Real Numbers have constitutive structures
> such as additive abelian group, multiplicative abelian group,
> topology generated by open intervals, totally ordered infinite set, and
> so on?
> At first one might think of forgetful functors, but then what would
> be the category in which Real Numbers is one object among many?
> Or, one might say take a category with exactly one object and a
> functor to each of the categories of the constitutive structures. This
> makes
> the Real Numbers look like an "element" of the "intersection" of
> diverse categories. Then the Complex Numbers or the Hyperreal Numbers
> which contain
> the Real Numbers as sub-objects in certain ways are "elements" of
> other "intersections" of categories. What am I talking about?
>
> Ellis D. Cooper
>
>
>
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