RE: Constitutive Structures

["F. William Lawvere" <wlawvere@hotmail.com>]

It seems that what Ellis is asking for is not so much the interesting richness per se
of the real numbers but "the proper categorical framework", that is a fragment of  objective logicto explain how we relate partial structures of the  "same thing". Not necessarily an "intersection"but more precisely
  an inverse limit of a diagram of forgetful functors 
may be the right sort of thing. Straining through many related layersvia naturality is the standard way to extract the Structure of a given functor measuring given mathematical objects. Can it dually be a way to extract a image of the objects themselves?Bill
> Date: Thu, 7 Apr 2011 08:50:11 -0400
> To: categories@mta.ca
> From: xtalv1@netropolis.net
> Subject: categories: Constitutive Structures
> 
> 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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