* Constitutive Structures @ 2011-04-07 12:50 Ellis D. Cooper 2011-04-08 10:18 ` Andrej Bauer ` (5 more replies) 0 siblings, 6 replies; 10+ messages in thread From: Ellis D. Cooper @ 2011-04-07 12:50 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-07 12:50 Constitutive Structures Ellis D. Cooper @ 2011-04-08 10:18 ` Andrej Bauer 2011-04-12 4:42 ` Andrej Bauer 2011-04-09 22:53 ` Vaughan Pratt ` (4 subsequent siblings) 5 siblings, 1 reply; 10+ messages in thread From: Andrej Bauer @ 2011-04-08 10:18 UTC (permalink / raw) To: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-08 10:18 ` Andrej Bauer @ 2011-04-12 4:42 ` Andrej Bauer 0 siblings, 0 replies; 10+ messages in thread From: Andrej Bauer @ 2011-04-12 4:42 UTC (permalink / raw) To: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-07 12:50 Constitutive Structures Ellis D. Cooper 2011-04-08 10:18 ` Andrej Bauer @ 2011-04-09 22:53 ` Vaughan Pratt 2011-04-13 19:24 ` F. William Lawvere ` (3 subsequent siblings) 5 siblings, 0 replies; 10+ messages in thread From: Vaughan Pratt @ 2011-04-09 22:53 UTC (permalink / raw) To: categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* RE: Constitutive Structures 2011-04-07 12:50 Constitutive Structures Ellis D. Cooper 2011-04-08 10:18 ` Andrej Bauer 2011-04-09 22:53 ` Vaughan Pratt @ 2011-04-13 19:24 ` F. William Lawvere 2011-04-14 23:11 ` Richard Garner ` (2 subsequent siblings) 5 siblings, 0 replies; 10+ messages in thread From: F. William Lawvere @ 2011-04-13 19:24 UTC (permalink / raw) To: xtalv1, categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-07 12:50 Constitutive Structures Ellis D. Cooper ` (2 preceding siblings ...) 2011-04-13 19:24 ` F. William Lawvere @ 2011-04-14 23:11 ` Richard Garner 2011-04-15 17:14 ` Prof. Peter Johnstone [not found] ` <alpine.LRH.2.00.1104151758260.15302@siskin.dpmms.cam.ac.uk> [not found] ` <BANLkTinFqZ+fKSqy3OCWbvGADKQGCO8yeA@mail.gmail.com> 5 siblings, 1 reply; 10+ messages in thread From: Richard Garner @ 2011-04-14 23:11 UTC (permalink / raw) To: Ellis D. Cooper; +Cc: categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-14 23:11 ` Richard Garner @ 2011-04-15 17:14 ` Prof. Peter Johnstone 0 siblings, 0 replies; 10+ messages in thread From: Prof. Peter Johnstone @ 2011-04-15 17:14 UTC (permalink / raw) To: Richard Garner; +Cc: Ellis D. Cooper, categories 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/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
[parent not found: <alpine.LRH.2.00.1104151758260.15302@siskin.dpmms.cam.ac.uk>]
* Re: Constitutive Structures [not found] ` <alpine.LRH.2.00.1104151758260.15302@siskin.dpmms.cam.ac.uk> @ 2011-04-16 0:31 ` Richard Garner 0 siblings, 0 replies; 10+ messages in thread From: Richard Garner @ 2011-04-16 0:31 UTC (permalink / raw) To: Prof. Peter Johnstone; +Cc: Ellis D. Cooper, categories 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
[parent not found: <BANLkTinFqZ+fKSqy3OCWbvGADKQGCO8yeA@mail.gmail.com>]
* Re: Constitutive Structures [not found] ` <BANLkTinFqZ+fKSqy3OCWbvGADKQGCO8yeA@mail.gmail.com> @ 2011-04-16 0:53 ` Richard Garner 2011-04-18 3:49 ` David Roberts 0 siblings, 1 reply; 10+ messages in thread From: Richard Garner @ 2011-04-16 0:53 UTC (permalink / raw) To: Prof. Peter Johnstone; +Cc: Ellis D. Cooper, categories 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
* Re: Constitutive Structures 2011-04-16 0:53 ` Richard Garner @ 2011-04-18 3:49 ` David Roberts 0 siblings, 0 replies; 10+ messages in thread From: David Roberts @ 2011-04-18 3:49 UTC (permalink / raw) To: categories 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 10+ messages in thread
end of thread, other threads:[~2011-04-18 3:49 UTC | newest] Thread overview: 10+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2011-04-07 12:50 Constitutive Structures Ellis D. Cooper 2011-04-08 10:18 ` Andrej Bauer 2011-04-12 4:42 ` Andrej Bauer 2011-04-09 22:53 ` Vaughan Pratt 2011-04-13 19:24 ` F. William Lawvere 2011-04-14 23:11 ` Richard Garner 2011-04-15 17:14 ` Prof. Peter Johnstone [not found] ` <alpine.LRH.2.00.1104151758260.15302@siskin.dpmms.cam.ac.uk> 2011-04-16 0:31 ` Richard Garner [not found] ` <BANLkTinFqZ+fKSqy3OCWbvGADKQGCO8yeA@mail.gmail.com> 2011-04-16 0:53 ` Richard Garner 2011-04-18 3:49 ` David Roberts
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