David Leduc <david.leduc6 <at> googlemail.com> writes:
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
Here is a basic and quite natural interpretation (if someone has not already pointed this out): One can have a n.t F => G iff F is less defined than G and on their common domain (which is just the domain of F) there is a natural transformation from F => \rst{F} G. Partial functors, of course, form a restriction category so they are naturally partial order enriched (by restriction). This 2-cell structure must simply respect this partial order ... This is certainly not the only possibility, unfortunately ... for example why not also allow partial natural transformations ... which are less defined than the functor. Here one does have to be a bit careful: a natural transformation must "know" the subcategory it is working with ... thus defining the natural transformation as a function on arrows (rather than just objects) is worthwhile adjustment (see MacLane page 19, Excercise 5). This then works too .... I hope this helps. -robin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target). I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows. Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D. I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof). Best Thoughts, David Yetter ________________________________________ From: Robin Cockett <robin@ucalgary.ca> Sent: Monday, March 16, 2015 6:12 PM To: Categories list Subject: categories: Partial functors .. David Leduc <david.leduc6 <at> googlemail.com> writes:
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
Here is a basic and quite natural interpretation (if someone has not already pointed this out): One can have a n.t F => G iff F is less defined than G and on their common domain (which is just the domain of F) there is a natural transformation from F => \rst{F} G. Partial functors, of course, form a restriction category so they are naturally partial order enriched (by restriction). This 2-cell structure must simply respect this partial order ... This is certainly not the only possibility, unfortunately ... for example why not also allow partial natural transformations ... which are less defined than the functor. Here one does have to be a bit careful: a natural transformation must "know" the subcategory it is working with ... thus defining the natural transformation as a function on arrows (rather than just objects) is worthwhile adjustment (see MacLane page 19, Excercise 5). This then works too .... I hope this helps. -robin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
It seems to me that this way of modelling partial functors via functors doesn't work too. Consider the following example: let C be a category with two objects, let's call them x0 and x1 and suppose that * C[x0,x1] = C[x1,x0] are singletons whose elements we will call i and j respectively that are one inverse to each other * C[x0,x0] = C[x1,x1] = N the monoid of natural numbers S is the full subcategory of C containing just the object x0, i.e. is the monoid of natural number N seen as a category. There's an obvious functor P : S -> N, which is an isomorphism, by definition. This gives us a *non trivial* partial functor, nonetheless if we take by N~ to be the category obtained by adding freely a zero object to N a functor P' : C -> N~ that sends the object x1 in the zero object of N~ should send also x0 in the zero object because x0 and x1 are isomorphic in C and functors preserve isomorphisms. Since N~ is obtained by adding freely a zero object this implies that P' should be a constant functor and it could be the only functor sending the objects of C not in S in the zero object. This example seems to show that the adding a zero object may not be a solution to the problem. If I correctly understood Robin Cockett's construction below it seems something like this The category of partial functor from C to D should be a subcategory of the comma-category (Cat,D) where: * objects are functors P : S -> D, where S is a subcategory of C * morphisms from an object P : S -> D to an object P' : S' -> D are pairs (I,a) where I : S -> S' is an embedding of (sub)categories and a : P -> P'?I is a natural transformation between these two functors in Cat[S,D]. Composition between functors (in order to get a 2-category) should be defined in the obvious way. Giorgio On Tue, Mar 17, 2015 at 03:04:54PM +0000, David Yetter wrote:
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows.
Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D.
I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof).
Best Thoughts, David Yetter ________________________________________ From: Robin Cockett <robin@ucalgary.ca> Sent: Monday, March 16, 2015 6:12 PM To: Categories list Subject: categories: Partial functors ..
David Leduc <david.leduc6 <at> googlemail.com> writes:
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
Here is a basic and quite natural interpretation (if someone has not already pointed this out):
One can have a n.t F => G iff F is less defined than G and on their common domain (which is just the domain of F) there is a natural transformation from F => \rst{F} G. Partial functors, of course, form a restriction category so they are naturally partial order enriched (by restriction). This 2-cell structure must simply respect this partial order ...
This is certainly not the only possibility, unfortunately ... for example why not also allow partial natural transformations ... which are less defined than the functor. Here one does have to be a bit careful: a natural transformation must "know" the subcategory it is working with ... thus defining the natural transformation as a function on arrows (rather than just objects) is worthwhile adjustment (see MacLane page 19, Excercise 5). This then works too ....
I hope this helps.
-robin
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Ah I did say that this is certainly not the only alternative!!!! David's suggestion also works and is interesting in its own right. Adding a zero to categories is a monad on the category of (small) categories which is (as far as I can see) is a perfectly good example of a partial map classifier. However it does not classify ALL subcategories: once f is taken to the zero map then all composites gfh must be taken to the zero map. This means that the subcategory of 4 (i.e. the linear order 0 -> 1 -> 2 -> 3 ) which consists of the identity maps and the map 0 -> 3 cannot be classified (as the middle map is taken to zero). The subcategories which CAN be classified are exactly those which are factor closed (i.e. fg in \X' \subseteq \X => f,g \in \X'). BTW: The original suggestion seen in this light also works: adding a disconnected object, \X +1, is also a partial map classifier. However, the subcategories classified must be connection closed (i.e. a disjoint sum component of the category). This may not have been quite the original objective .... Of course, Cat does not have a partial map classifier for ALL subcategories ... however, this does not stop one from building partial functor categories using all partial functors which have perfectly happy natural transformations :-) ... but there are still some choices to make along the natural transformation road. All choices lead to interesting alternatives ... some more interesting than others! -robin On Tue, Mar 17, 2015 at 9:04 AM, David Yetter <dyetter@ksu.edu> wrote:
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows.
Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D.
I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof).
Best Thoughts, David Yetter ________________________________________ From: Robin Cockett <robin@ucalgary.ca> Sent: Monday, March 16, 2015 6:12 PM To: Categories list Subject: categories: Partial functors ..
David Leduc <david.leduc6 <at> googlemail.com> writes:
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
Here is a basic and quite natural interpretation (if someone has not already pointed this out):
One can have a n.t F => G iff F is less defined than G and on their common domain (which is just the domain of F) there is a natural transformation from F => \rst{F} G. Partial functors, of course, form a restriction category so they are naturally partial order enriched (by restriction). This 2-cell structure must simply respect this partial order ...
This is certainly not the only possibility, unfortunately ... for example why not also allow partial natural transformations ... which are less defined than the functor. Here one does have to be a bit careful: a natural transformation must "know" the subcategory it is working with ... thus defining the natural transformation as a function on arrows (rather than just objects) is worthwhile adjustment (see MacLane page 19, Excercise 5). This then works too ....
I hope this helps.
-robin
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
--001a11c133443cbef6051181db4e Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable <div dir=3D"ltr">Ah I did say that this is certainly not the only alternati= ve!!!!<div><br></div><div>David's suggestion also works and is interest= ing in its own right. =C2=A0</div><div><br></div><div>Adding a zero to cate= gories is a monad on the category of (small) categories which is (as far as= I can see) is a perfectly good example of a partial map classifier.=C2=A0 = However it does not classify ALL subcategories: once f is taken to the zero= map then all composites gfh must be taken to the zero map. =C2=A0 This mea= ns that the subcategory of 4 (i.e. the linear order 0 -> 1 -> 2 ->= 3 ) which consists of the identity maps and the map 0 -> 3 cannot be cl= assified (as the middle map is taken to zero).=C2=A0 The subcategories whic= h CAN be classified are exactly those which are factor closed (i.e. fg in \= X' \subseteq \X =3D> f,g \in \X'). =C2=A0=C2=A0</div><div><br></= div><div>BTW: The original suggestion seen in this light also works: adding= a disconnected object, \X +1, is also a partial map classifier.=C2=A0 Howe= ver, the subcategories classified must be connection closed =C2=A0(i.e. a d= isjoint sum component of the category).=C2=A0 This may not have been quite = the original objective ....</div><div><br></div><div>Of course, Cat does no= t have a partial map classifier for ALL subcategories ... however, this doe= s not stop one from building partial functor categories using all partial f= unctors which have perfectly happy natural transformations :-) ... but ther= e are still some choices to make along the natural transformation road.=C2= =A0 All choices lead to interesting alternatives ... some more interesting = than others!</div><div><br></div><div>-robin</div></div><div class=3D"gmail= _extra"><br><div class=3D"gmail_quote">On Tue, Mar 17, 2015 at 9:04 AM, Dav= id Yetter <span dir=3D"ltr"><<a href=3D"mailto:dyetter@ksu.edu" target= =3D"_blank">dyetter@ksu.edu</a>></span> wrote:<br><blockquote class=3D"g= mail_quote" style=3D"margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-l= eft:1ex">The previous suggestion of considering functors to D + 1 was a fal= se start for reasons Fred and Uwe pointed out, but it suggests a better app= roach:=C2=A0 consider functors to the category D~ formed from D by freely a= djoining a zero object.=C2=A0 Arrows not in S now have somewhere to go (the= zero arrow with the appropriate source and target).<br> <br> I think at the one-categorical level, taking Hom(C,D) to be the zero-preser= ving functors from C~ to D~, and letting C and D range over all small categ= ories gives=C2=A0 a category isomorphic to that of small categories with pa= rtial=C2=A0 functors as arrows.<br> <br> Natural transformations between (zero-preserving) functors from C~ to D~ wo= uld<br> then give a reasonable notion of partial natural transformations.=C2=A0 It = certainly captures some, at least, of the natural transformations "mor= e partial"=C2=A0 than their source functor, since there will be a zero= natural transformation between any two partial functors, corresponding to = a "defined nowhere" partial natural transformation when zero-ness= is interpreted as undefined as=C2=A0 it was in the correspondence between = zero-preserving functors from C~ to D~ and partial functors from C to D.<br=
<br> I'm not sure how this fits with the restrictions Robin points out.=C2= =A0 It seems to allow more partial natural transformations than Robin's= observation, since zero arrows can fill in whenever the image object under= either the source or target functor is undefined, a partial natural transf= ormation to be a natural transformation between the restrictions of the two= partial functors to the intersections of their domain of definition (or a = subcategory thereof).<br> <br> Best Thoughts,<br> David Yetter<br> ________________________________________<br> From: Robin Cockett <<a href=3D"mailto:robin@ucalgary.ca">robin@ucalgary= .ca</a>><br> Sent: Monday, March 16, 2015 6:12 PM<br> To: Categories list<br> Subject: categories: Partial functors ..<br> <br> David Leduc <david.leduc6 <at> <a href=3D"http://googlemail.com" t= arget=3D"_blank">googlemail.com</a>> writes:<br> <br> > A partial functor from C to D is given by a subcategory S of C and a<b= r> > functor from S to D. What is the appropriate notion of natural<br> > transformation between partial functors that would allow to turn small= <br> > categories, partial functors and those "natural transformations&q= uot; into<br> > a bicategory? The difficulty is that two partial functors from C to D<= br> > might not have the same definition domain.<br> <br> <br> Here is a basic and quite natural interpretation (if someone has not<br> already pointed this out):<br> <br> One can have a n.t=C2=A0 F =3D> G iff F is less defined than G and on th= eir common<br> domain (which is just the domain of F) there is a natural transformation<br=
from F =3D> \rst{F} G.=C2=A0 =C2=A0Partial functors, of course, form a r= estriction<br> category so they are naturally partial order enriched (by restriction).<br> This 2-cell structure must simply respect this partial order ...<br> <br> This is certainly not the only possibility, unfortunately ... for example<b= r> why not also allow partial natural transformations ... which are less<br> defined than the functor.=C2=A0 =C2=A0Here one does have to be a bit carefu= l: a<br> natural transformation must "know" the subcategory it is working = with ...<br> thus defining the natural transformation as a function on arrows (rather<br=
than just objects) is worthwhile adjustment (see MacLane page 19, Excercise= <br> 5).=C2=A0 This then works too ....<br> <br> I hope this helps.<br> <br> -robin<br> <br> <br> <br> [For admin and other information see: <a href=3D"http://www.mta.ca/~cat-dis= t/" target=3D"_blank">http://www.mta.ca/~cat-dist/</a> ]<br> <br> </blockquote></div><br></div> --001a11c133443cbef6051181db4e-- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear David, of recent papers, there are for exemple K. Dosen, Z. Petric Coherence in substructural categories, Studia Logica, vol.
70 (2002), pp. 271-296 (available at: http://arXiv.org)
where the diagonal case (Relevant categories) is treated separately. The cartesian case is treated in
K. Dosen, Z. Petric, The maximality of cartesian categories, Mathematical logic Quarterly, vol. 47 (2001), pp. 137-144 (available at: http://arXiv.org)
and also in our book Proof-Theoretical Coherence, Chapter 9. (available at http://www.mi.sanu.ac.rs/~kosta/coh.pdf).
There is also a recent paper
K. Dosen, On Sets of Premises (available at: http://arXiv.org) As to the historical aspect, there was a theorem in an old paper by Grigori Mints, published in Russian in 1980 (reprinted in "Selected papers in Proof Theory", Bibliopolis, 1992), but I have to find the paper and exact formulation of his theorem. regards, Sergei Soloviev Le Mardi 17 Mars 2015 16:04 CET, David Yetter <dyetter@ksu.edu> a écrit:
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows.
Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D.
I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof).
Best Thoughts, David Yetter ________________________________________ From: Robin Cockett <robin@ucalgary.ca> Sent: Monday, March 16, 2015 6:12 PM To: Categories list Subject: categories: Partial functors ..
David Leduc <david.leduc6 <at> googlemail.com> writes:
A partial functor from C to D is given by a subcategory S of C and a functor from S to D. What is the appropriate notion of natural transformation between partial functors that would allow to turn small categories, partial functors and those "natural transformations" into a bicategory? The difficulty is that two partial functors from C to D might not have the same definition domain.
Here is a basic and quite natural interpretation (if someone has not
already pointed this out):
One can have a n.t F => G iff F is less defined than G and on their common domain (which is just the domain of F) there is a natural transformation from F => \rst{F} G. Partial functors, of course, form a restriction category so they are naturally partial order enriched (by restriction). This 2-cell structure must simply respect this partial order ...
This is certainly not the only possibility, unfortunately ... for example why not also allow partial natural transformations ... which are less defined than the functor. Here one does have to be a bit careful: a natural transformation must "know" the subcategory it is working with ... thus defining the natural transformation as a function on arrows (rather than just objects) is worthwhile adjustment (see MacLane page 19, Excercise 5). This then works too ....
I hope this helps.
-robin
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear All David Leduc’s statement: "A partial functor from C to D is given by a subcategory S of C and a functor from S to D.” If this is taken as given then today I don’t want to add to what people have said. But there is another notion of partial functor that I learnt from Brian Day who learnt it from Bill Lawvere. If we are to replace 2 in Set by Set in Cat, then characteristic maps C —> 2 should become presheaves C^op —> Set, so injective functions S—> C should become discrete fibrations E —> C. Then a partial map from C to D would be a span C <— E —> D with C <— E a discrete fibration and E —> D any functor. The partial map classifier is then the coproduct completion FamD of D. Add size restrictions, to taste. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
One had better take care to note explicitly how to compose maps to and from the newly adjoined "zero object", if following Yetter's idea,
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
For, given a and b objects in D and writing z for the newly adjoined zero object, what are we to take for compositions a --> * --> b ? Or were we also to adjoin "zero maps" z_(a,b) to each existing homset D(a, b), and then set these compositions all equal to those new zero maps? That suggests Yetter really meant to propose forming the free *pointed* category with zero object freely engendered by D ... or maybe that's what his words already meant to convey? Apologies if I was deaf to that tone :-) . Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Ross, I think you meant the *product* completion Fam(D*)*. If you use the coproduct completion, then you need to use discrete *opfibrations* and covariant Set-valued functors. There are some details in the appendix to our joint paper “The formal theory of monads II”. Steve.
On 18 Mar 2015, at 11:47 am, Ross Street <ross.street@mq.edu.au> wrote:
Dear All
David Leduc’s statement: "A partial functor from C to D is given by a subcategory S of C and a functor from S to D.”
If this is taken as given then today I don’t want to add to what people have said.
But there is another notion of partial functor that I learnt from Brian Day who learnt it from Bill Lawvere. If we are to replace 2 in Set by Set in Cat, then characteristic maps C —> 2 should become presheaves C^op —> Set, so injective functions S—> C should become discrete fibrations E —> C. Then a partial map from C to D would be a span C <— E —> D with C <— E a discrete fibration and E —> D any functor. The partial map classifier is then the coproduct completion FamD of D. Add size restrictions, to taste.
Ross
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I don't this this works: if (g .g') is in the domain of the partial functor and g is not, then g will be sent to 0 by the extension of the partial functor, hence g.g' will also be sent to 0 which shouldn't be the case because it is in the domain of the functor. Also it is not entirely clear to me if the original question wanted functor defined on a full subcategory or on an arbitrary sub-category. But in both case, my opinion would be that the only reasonable notion is to talk about "partial natural transformations" which are defined on a subcategory included in the domain of definition of both functor (and hence are natural transformation between ordinary functor). Best wishes, Simon Henry.
The previous suggestion of considering functors to D + 1 was a false start for reasons Fred and Uwe pointed out, but it suggests a better approach: consider functors to the category D~ formed from D by freely adjoining a zero object. Arrows not in S now have somewhere to go (the zero arrow with the appropriate source and target).
I think at the one-categorical level, taking Hom(C,D) to be the zero-preserving functors from C~ to D~, and letting C and D range over all small categories gives a category isomorphic to that of small categories with partial functors as arrows.
Natural transformations between (zero-preserving) functors from C~ to D~ would then give a reasonable notion of partial natural transformations. It certainly captures some, at least, of the natural transformations "more partial" than their source functor, since there will be a zero natural transformation between any two partial functors, corresponding to a "defined nowhere" partial natural transformation when zero-ness is interpreted as undefined as it was in the correspondence between zero-preserving functors from C~ to D~ and partial functors from C to D.
I'm not sure how this fits with the restrictions Robin points out. It seems to allow more partial natural transformations than Robin's observation, since zero arrows can fill in whenever the image object under either the source or target functor is undefined, a partial natural transformation to be a natural transformation between the restrictions of the two partial functors to the intersections of their domain of definition (or a subcategory thereof).
Best Thoughts, David Yetter
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (8)
-
David Yetter -
Fred E.J. Linton -
Giorgio Mossa -
henry＠phare.normalesup.org -
Robin Cockett -
Ross Street -
Sergei Soloviev -
Steve Lack