Re: Constitutive Structures

[Andrej Bauer <andrej.bauer@andrej.com>]

You may wish to look at Davorin Lešnik's Ph.D. thesis, where he
studies real numbers in a constructive setting (without choice). He
identifies  suitable categories inside of which the real numbers exist
as an object with a universal property that determines the reals up to
isomorphism. The various categories correspond to the various
substructure of the reals (order, additive group, ring, etc.)

An interesting question is where to find his Ph.D. thesis. I will make
him publish it somewhere on the web and will come back to you with a
link.

With kind regards,

Andrej

On Thu, Apr 7, 2011 at 2:50 PM, Ellis D. Cooper <xtalv1@netropolis.net> 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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