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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