Phil Scott | 1 Jul 03:15 2004
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Re: Questions on dinatural transformations.

In general, the naive horizontal merging of dinaturals fails to be
dinatural.  This is discussed in the article "Functorial
Polymorphism" by Bainbridge, Freyd, Scedrov and me (Theoretical Computer
Science 1990, pp. 35-64).

Several counterexamples  are given there.

For example, in a cartesian closed category of domains or CPO's, consider
a dinatural family Y_A:  A^A --> A (e.g. in domains, let Y_A = the least
fixed point operator).  If you were able to compose this with the
"polymorphic identity" dinat id_A: 1---> A^A (i.e. a dinat from
constant functor 1 to (-) ==> (-)  where  id_A = the transpose of the
identity on A),  then the category would be degenerate (proved in BFSS,
Appendix A.4).

Of course, if the middle diamond (of an attempted merging of two dinat
families) is a pullback or pushout, then merging works. (see BFSS, Fact
1.2).

Re vertical merging, some things can be said quite generally: e.g BFSS,
Propn. 1.3.

For various generalizations, see Peter Freyd's paper "Structural
Polymorphism" (in TCS, 1993, pp.107-129).  Soloviev has also discussed
compositionality of dinats in several articles in JPAA.

                    Philip Scott

On Tue, 29 Jun 2004, Noson Yanofsky wrote:

(Continue reading)

Robin Houston | 1 Jul 12:12 2004
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Re: Grothendieck bio?

On Wed, Jun 30, 2004 at 10:43:59AM -0700, Galchin Vasili wrote:
>    Does anybody know of a Grothendieck biography?

This question came up recently on sci.math.research, where there were
several interesting responses. The thread is at

  http://groups.google.com/groups?threadm=0t3t3bw3rrcd%40legacy

Robin

Vaughan Pratt | 1 Jul 19:01 2004
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Re: Questions on dinatural transformations.


>From: noson <at> sci.brooklyn.cuny.edu
>I was hoping that the category of small categories, functors and
>dinat transformations...

There's a category problem already at this point.  Dinats don't go between
functors F,G:C->D, they go between sesquifunctors F:C^op x C->D and differ
from n.t.'s of that type by only being defined on the diagonal of C^op x C.
The off-diagonal and non-identity-morphism entries in F,G only participate
in the dinaturality condition, not in the transformation itself.

>a) It is well known that there is no vertical
>composition of dinatural transformations.
>How about horizontal composition?

Before you can compose dinats horizontally you have to be able to compose
the sesquifunctors they bridge.  I don't know how others do this, but if I
had to compose G:D^op x D -> E with F:C^op x C -> D, my inclination would
be to restrict the evident composite G(F(a,b),F(c,d)) to a=d, b=c (i.e.
where the variances match up).  That is, GoF:C^op x C -> E is defined by
G(F(c,c'),F(c',c)) on object pairs (c',c) of C^op x C, with the expected
extension to morphism pairs (f',f) where f':c'->d' in C^op (i.e.
f':d'->c' in C) and f:c->d in C, namely

  G(F(f,f'),F(f',f)): G(F(c,c'),F(c',c)) -> G(F(d,d'),F(d',d)).

With that (or some) choice of sesquifunctor composition one can then ask
about horizontal composition tos where s:F->F', t:G->G'.  How would you
whisker a dinatural on the left, i.e. apply the whisker G:D^op x D->E
on the left to the dinat s:F->F' on the right where F,F':C^op x C->D?
(Continue reading)

Robert Knighten | 2 Jul 02:23 2004

Grothendieck bio?

Galchin Vasili writes:
 > Hello,
 >
 >    Does anybody know of a Grothendieck biography? For
 > me many times it is helpful to read a bio to the
 > historical development of a person's work.
 >
 > Thanks in advance, Bill Halchin

I'm pretty sure there is no book length biography, but there is some
information at:

http://www.math.jussieu.fr/~leila/biographic.php

which is really the information from www.grothendieck-circle.org which appears
to be off-line at the moment.

-- Bob

--
Robert L. Knighten
Robert <at> Knighten.org

Ronnie Brown | 2 Jul 15:32 2004
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Re: Grothendieck bio?

People may like to look at the Spectator article:

http://www.lewrockwell.com/spectator/spec262.html

I thought Recollte et Semaille translated as `Reaping and sowing'?

He was a great correspondent in the 1980s, and we have a mass of letters
which will be transferred to an archive in good hands in due course (my
administrative matters have got too much at present). This correspondence
is relevant to Pursuing Stacks, which was written in English as a response
to our correspondence in English, which he describes in `Esquisse d'un
programme' (EdP) as `a baton rompu' (ranging over this and that). In fact
EdP is available in French and English translation in books by Pierre
Lochack and Leila Schneps (LMS Lecture notes series). Pursuing Stacks was
to have been volume 1 of a series on `The long march towards Galois
theory', written in a new informal style, as in Pursuing Stacks, where the
thought is open to view. I suspect that writing R&S distracted him from
this aim.

I expect to explain the correspondence which led to Pursuing Stacks being
sent to me, and Larry Breen, and then circulated from Bangor,  in 1983, but
have not yet got round to it.

In any case, Pursuing Stacks seems to be increasingly influential, as does
EdP.

Ronnie Brown

Robin Houston wrote:
>
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Claudio Hermida | 3 Jul 02:20 2004
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Re: Questions on dinatural transformations.

Vaughan Pratt wrote:

>>From: noson <at> sci.brooklyn.cuny.edu
>>I was hoping that the category of small categories, functors and
>>dinat transformations...
>>
>>
>
>There's a category problem already at this point.  Dinats don't go between
>functors F,G:C->D, they go between sesquifunctors F:C^op x C->D and differ
>from n.t.'s of that type by only being defined on the diagonal of C^op x C.
>The off-diagonal and non-identity-morphism entries in F,G only participate
>in the dinaturality condition, not in the transformation itself.
>
>
>
>>a) It is well known that there is no vertical
>>composition of dinatural transformations.
>>How about horizontal composition?
>>
>>
>
>Before you can compose dinats horizontally you have to be able to compose
>the sesquifunctors they bridge.  I don't know how others do this, but if I
>had to compose G:D^op x D -> E with F:C^op x C -> D, my inclination would
>be to restrict the evident composite G(F(a,b),F(c,d)) to a=d, b=c (i.e.
>where the variances match up).  That is, GoF:C^op x C -> E is defined by
>G(F(c,c'),F(c',c)) on object pairs (c',c) of C^op x C, with the expected
>extension to morphism pairs (f',f) where f':c'->d' in C^op (i.e.
>f':d'->c' in C) and f:c->d in C, namely
(Continue reading)

Colin McLarty | 3 Jul 19:45 2004
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Quick surveys of categorical logic

For a philosophy article I need one or two comprehensive references to
categorical logic, especially emphasizing the array of doctrines.  I mean,
from algebraic theories to left exact theories to coherent theories and such.

I have hundreds of references.  I need one or two.  Very quick surveys are
fine for this, even preferable.  What is there?

best, Colin

Robert Seely | 3 Jul 23:29 2004
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Re: Quick surveys of categorical logic

An old but still useful reference is Kock & Reyes' article in the Handbook of
Mathematical Logic (of course, for a mid-70's paper, it's a bit dated by
now!).  Phil Scott has written some more recent ones, including "Some
Aspects of Categories in Computer Science (Survey Article on Categorical
Logic)" and "Category Theory for Linear Logicians" (with R. Blute),
available on his website (he can provide the pub refs).

-= rags =-

On Sat, 3 Jul 2004, Colin McLarty wrote:

> For a philosophy article I need one or two comprehensive references to
> categorical logic, especially emphasizing the array of doctrines.  I mean,
> from algebraic theories to left exact theories to coherent theories and such.
>
> I have hundreds of references.  I need one or two.  Very quick surveys are
> fine for this, even preferable.  What is there?
>
> best, Colin
>
>
>

--
<rags <at> math.mcgill.ca>
<www.math.mcgill.ca/rags>

Max Kelly | 5 Jul 08:35 2004
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Dinatural transformations

Nason Yanofsky asked a question about composition of dinatural
transformations, and there have been a number of replies - but none
giving the reference I should have expected, namely [S.Eilenberg and
G.M.Kelly, A generalization of the functorial calculus,, J.Algebra 3
(1966), 366 - 375] .

What Sammy and I  were concerned with were such families as  the
evaluationn e_A,B : [A,B] o A --> B , where o is a tensor product and [
, ] is an internal hom. Here e_A,B is natural in B in the usual sense;
it is also natural, in our extended sense, in the variable A, which
appears twice on one side of the arrow, but with two opposite variances.
Similar for d_A,B : A --> [B, AoB]. In certain circumstances one can
compose such "natural transformations" to form new ones:one example is
the composite

                  AoB ---------> [B, AoB] o B ----------> AoB
                             d_A,B o B                     e_B, AoB

which is in fact the identity natural transformation.

Sammy and I gave a general treatment in the article above. Later, others
generalised our "extended naturals" to get the notion of dinatural
transformation. Since these do not compose except in some very special
cases, I find them to be of limited interest.   In contrast, I find that
I still use the Eilenberg-Kellycalculus from time to time.

Max Kelly.

Logical Methods in CS | 5 Jul 12:24 2004

New Journal

NEW JOURNAL----NEW JOURNAL----NEW JOURNAL----NEW JOURNAL----NEW JOURNAL

-------------------------EXCUSE MULTIPLE COPIES-----------------------

L O G I C A L   M E T H O D S    I N   C O M P U T E R   S C I E N C E

Dear Colleague:

We are writing to inform you about a new open-access, online, refereed
journal: "Logical Methods in Computer Science". As an open-access
publication, the journal will be freely available on the web. This new
journal will be devoted to all theoretical and practical topics in
computer science related to logic in a broad sense.  You can find the
homepage at

        http://www.lmcs-online.org

The journal will open to submissions on September 1, 2004.

It will be published under the auspices of The International Federation
for Computational Logic: http://www.colognet.org/IFCoLog/.
The journal will technically be published as an overlay of the Computing
Research Repository (CoRR), see http://arxiv.org/archive/cs/intro.html.

On the homepage you find a flier and a leaflet containing the basic
information about the new journal. We would appreciate your posting
and distributing the information, and encouraging potential authors to
submit to Logical Methods in Computer Science.

You may have heard about the various developments in the past couple of
(Continue reading)


Gmane