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/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
-
Andrej Bauer -
Ellis D. Cooper -
F. William Lawvere -
Prof. Peter Johnstone -
Richard Garner -
Vaughan Pratt