Re: Constitutive Structures

[Richard Garner <richard.garner@mq.edu.au>]

Here's a possible answer using toposes. I don't really know enough
topos theory to do this properly so I will be busking it a bit;
hopefully someone more knowledgeable than I can tell me what I am up
to! We define a factorisation system (E,M) on the 2-category of
Grothendieck toposes, generated by the following M-maps. For each n,
we take the obvious geometric morphism from the classifying topos of
an object equipped with an n-ary relation to the object classifier;
and we take that geometric morphism from the object classifier to the
classifying topos of a monomorphism which classifies the identity map
on the generic object. With any luck this generates a factorisation
system on GTop; with equal luck it is a well-known one, but my
knowledge of the taxonomy of classes of geometric morphisms is
sufficiently hazy that I cannot say which it might be. In any case,
the hope is that M-maps into the object classifier should correspond
to single-sorted geometric theories. Now we work in the category of
such M-maps into Set[O], and in there, there is an object which
represents all the constitutive substructures of the reals. The object
in question is obtained as the M-part of the (E,M) factorisation of
the geometric morphism Set -> Set[O] which classifies the real
numbers; it is the "complete theory of the reals", but not with
respect to any particular structure, but rather with respect to all
possible structures (within geometric logic) that we might impose on
it. Unfortunately this would not capture, e.g., the second-order
structures we might impose on the reals, but it's a start.

(Of course, if we were merely interested in structures expressible by
finitary algebraic theories, then we could consider the category of
finitary monads on Set, and in there, the finitary coreflection of the
codensity monad of the reals. That was my initial reaction to this
problem, and the above is supposed to generalise this in some sense).

Richard


On 7 April 2011 22:50, 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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