Constitutive Structures

Constitutive Structures

[Ellis D. Cooper]

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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Re: Constitutive Structures

[Andrej Bauer]

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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Re: Constitutive Structures

[Vaughan Pratt]

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
>
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


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Re: Constitutive Structures

[Andrej Bauer]

The promised URL for Davorin's thesis "Synthetic Topology and Constructive
Metric Spaces" is now available at

http://www.fmf.uni-lj.si/storage/19160/PhD_Davorin.pdf

Chapter 3 is devoted to an excruciatingly detailed treatment of real
numbers. It may give you some ideas on how to get your structures
working in a similar way.

Other cool things in the thesis are a constructive Urysohn space, and
a notion of co-dominance with a symmetric treatment of open and closed
sets in constructive topology.

With kind regards,

Andrej


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RE: Constitutive Structures

[F. William Lawvere]

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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Re: Constitutive Structures

[Richard Garner]

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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Re: Constitutive Structures

[Prof. Peter Johnstone]

Dear Richard,

That's an ingenious idea, but I don't think it helps. The
factorization system is indeed a well-known one: it's the
hyperconnected--localic factorization [proof below], and
it is indeed true that M-maps into Set[O] correspond to
single-sorted geometric theories (Elephant, D3.2.5). But
every morphism Set --> Set[O] (in particular the one which
classifies the real numbers) is localic, so you just end up
with the topos of sets.

Here's the proof. The morphisms you describe are all localic,
so it's enough to prove that any morphism orthogonal to them all
is hyperconnected. But orthogonality to the last morphism you
list, for a morphism f: F --> E, says precisely that if m is
a mono in E and f^*(m) is iso then m is iso, i.e. that f is
surjective. Then orthogonality to the first group (actually
you only need the case n=1) says that f^* is `full on subobjects',
i.e. that every subobject of f^*(A) is of the form f^*(B) for a
unique (up to isomorphism) B >--> A. Applying this to the graphs
of morphisms, you get that f^* is full in the usual sense;
applying it to arbitrary subobjects, you get the criterion for
hyperconnectedness given in Elephant, A4.6.6(ii).

Peter Johnstone

On Fri, 15 Apr 2011, Richard Garner wrote:

> 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
>

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Re: Constitutive Structures

[Richard Garner]

Thanks Peter. It did occur to me last night that this probably was the
hyperconnected-localic factorisation and it is nice to have this
feeling confirmed! The problem is that the factorisation system I
described allows one to adjoin n-ary relations to _arbitrary_ objects
of Set[O], rather than merely to the generic object. In particular, as
you point out, of the first group of maps I listed it is only
necessary to consider the case n=1, and in fact on looking at at your
proof, orthogonality to this immediately implies orthogonality to the
last of the maps I listed.

Here's an attempt to overcome this; I suspect it will end up suffering
the same fate as the previous one but you never know! Rather than
describing a factorisation system on GTop, I am going to describe one
on GTop / Set[O]. The generating right maps will simply be the maps
from the classifying topos of an object equipped with an n-ary
relation into Set[O], though now these maps are viewed as maps over
Set[O]. If this generates a factorisation system (E, M), then its
M-maps with codomain E --> Set[O] will correspond to those things
constructible by repeatedly adjoining n-ary relations or equations
between n-ary relations to the specified object of E. Every such map
will be localic, but I think that the E-maps are no longer the
hyperconnected morphisms; the inverse image part of such a map need
only be full on subobjects of the specified object of its domain.

Now on factorising the unique map from R: Set --> Set[O] into the
terminal object of GTop / Set[O], it is possible that we obtain
something non-trivial which captures the structures (in geometric
logic) supported by the reals. I am however a bit hesitant about this
as my feeling is that if p: E --> F is an E-map of toposes over
Set[O], and F --> Set[O] is localic, then p probably is actually
hyperconnected (i.e., fullness on subobjects of the (image of) the
generic object implies fullness on all subobjects) so that we are back
in the situation we were in before...

Richard

On 16 April 2011 03:14, Prof. Peter Johnstone
<P.T.Johnstone@dpmms.cam.ac.uk> wrote:
> Dear Richard,
>
> That's an ingenious idea, but I don't think it helps. The
> factorization system is indeed a well-known one: it's the
> hyperconnected--localic factorization [proof below], and
> it is indeed true that M-maps into Set[O] correspond to
> single-sorted geometric theories (Elephant, D3.2.5). But
> every morphism Set --> Set[O] (in particular the one which
> classifies the real numbers) is localic, so you just end up
> with the topos of sets.
>
> Here's the proof. The morphisms you describe are all localic,
> so it's enough to prove that any morphism orthogonal to them all
> is hyperconnected. But orthogonality to the last morphism you
> list, for a morphism f: F --> E, says precisely that if m is
> a mono in E and f^*(m) is iso then m is iso, i.e. that f is
> surjective. Then orthogonality to the first group (actually
> you only need the case n=1) says that f^* is `full on subobjects',
> i.e. that every subobject of f^*(A) is of the form f^*(B) for a
> unique (up to isomorphism) B >--> A. Applying this to the graphs
> of morphisms, you get that f^* is full in the usual sense;
> applying it to arbitrary subobjects, you get the criterion for
> hyperconnectedness given in Elephant, A4.6.6(ii).
>
> Peter Johnstone
>
> On Fri, 15 Apr 2011, Richard Garner wrote:
>
>> 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

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Re: Constitutive Structures

[Richard Garner]

In fact, I think the condition of being an E-map in this new sense
says: f : E --> F over Set[O] is such a map just when, on taking the
hyperconnected-localic factorisations E --> E' --> Set[O] and F --> F'
--> Set[O], the induced geometric morphism f' : E' --> F' is
hyperconnected. In particular, if F --> Set[O] is localic, then f is
an E-map if and only if it is hyperconnected. So this gets us no
further than before. Oh well!

Richard

On 16 April 2011 10:31, Richard Garner <richard.garner@mq.edu.au> wrote:
> Thanks Peter. It did occur to me last night that this probably was the
> hyperconnected-localic factorisation and it is nice to have this
> feeling confirmed! The problem is that the factorisation system I
> described allows one to adjoin n-ary relations to _arbitrary_ objects
> of Set[O], rather than merely to the generic object. In particular, as
> you point out, of the first group of maps I listed it is only
> necessary to consider the case n=1, and in fact on looking at at your
> proof, orthogonality to this immediately implies orthogonality to the
> last of the maps I listed.
>
> Here's an attempt to overcome this; I suspect it will end up suffering
> the same fate as the previous one but you never know! Rather than
> describing a factorisation system on GTop, I am going to describe one
> on GTop / Set[O]. The generating right maps will simply be the maps
> from the classifying topos of an object equipped with an n-ary
> relation into Set[O], though now these maps are viewed as maps over
> Set[O]. If this generates a factorisation system (E, M), then its
> M-maps with codomain E --> Set[O] will correspond to those things
> constructible by repeatedly adjoining n-ary relations or equations
> between n-ary relations to the specified object of E. Every such map
> will be localic, but I think that the E-maps are no longer the
> hyperconnected morphisms; the inverse image part of such a map need
> only be full on subobjects of the specified object of its domain.
>
> Now on factorising the unique map from R: Set --> Set[O] into the
> terminal object of GTop / Set[O], it is possible that we obtain
> something non-trivial which captures the structures (in geometric
> logic) supported by the reals. I am however a bit hesitant about this
> as my feeling is that if p: E --> F is an E-map of toposes over
> Set[O], and F --> Set[O] is localic, then p probably is actually
> hyperconnected (i.e., fullness on subobjects of the (image of) the
> generic object implies fullness on all subobjects) so that we are back
> in the situation we were in before...
>
> Richard
>

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Re: Constitutive Structures

[David Roberts]

Hi all,

As well as the real numbers, one could consider also the example of
the free \lambda-ring, the ring of symmetric polynomials on a
countable number of indeterminates. This has a large number of 'faces'
that present themselves in many different ways.

David


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