Re: exploiting similarities and analogies

Re: exploiting similarities and analogies

[Vaughan Pratt]

Al Vilcius wrote:
> Has anyone explored, either formally or informally, the connection between
> the Melzak Bypass Principle (MBP) and adjoints?

Considering that Google returned 'Your search - "Melzak Bypass
Principle" - did not match any documents,' the acronym might be a tad
premature.  But in the spirit of the question the connection I'd be
tempted to make (predictably given my biases) would be with duality
rather than adjunction.  Thus floor: R --> Z as right adjoint to the
inclusion of Z into R is a (posetal) example of adjunction (with the
numerical inequalities x <= y as the morphisms) that in this case
forgets the continuum structure of R, whereas duality being an
involution (at least up to equivalence) necessarily retains all the
categorical structure in mirror image.

For example to understand finite distributive lattices, transform them
to finite posets, work on them in that guise, and transform back at the
end.  Being a categorical duality, the resulting understanding of
distributive lattices includes their homomorphisms, which under this
duality become monotone functions.  Complete atomic Boolean algebras are
even simpler: viewed in the duality mirror they are just ordinary sets
transforming by functions.  Hence the homset CABA(2^X,2^Y) of complete
Boolean homomorphisms from 2^X to 2^Y as complete atomic Boolean
algebras is (representable as) the set X^Y of functions f:X --> Y  (so
for example there are 5^3 = 125 Boolean homomorphisms from 2^5 to 2^3).
  Likewise one can understand Boolean algebras and their homomorphisms
in terms of their dual Stone spaces and continuous functions, while the
self-duality of any category of finite-dimensional vector spaces over a
given field is a linchpin of matrix algebra and essential to the linear
algebra examples you cited for "MBP."

There is such a diversity of fruitful dualities, of widely varying
characters, that one despairs of finding any uniformity to them.  For me
this is where Chu spaces enter.  What I find so appealing about Chu
spaces is that they tap into a vein of uniformity running through these
disparate examples whose global structure is that of *-autonomous
categories, or linear logic when seen "in the light of logic."  (One can
view *-autonomous categories as being to *-autonomous categories of Chu
spaces roughly as Boolean algebras are to fields of sets and toposes to
toposes of presheaves.)  All of the dualities listed above and many more
can be exhibited as (usually not *-autonomous) subcategories of
Chu(Set,K) for a suitable set K, with each such category and its dual
connected by the self-duality of the *-autonomous category Chu(Set,K)
itself.

http://www.tac.mta.ca/tac/index.html#vol17 , the special issue of TAC on
Chu spaces that Valeria de Paiva and I edited, conveys some of the
flavor of this.  Mike Barr gives a history of Chu spaces at
http://www.tac.mta.ca/tac/volumes/17/1/17-01.pdf, while the preface at
http://www.tac.mta.ca/tac/volumes/17/pref/17-pref.pdf supplements Mike's
history and gives an overview of the papers in the volume.  My 1997
Coimbra notes http://boole.stanford.edu/pub/coimbra.pdf on Chu spaces
play a more introductory role (the first half anyway, the second half
emphasized linear logic more than I would have if I were writing it today).

The Chu space scene has been a bit quiet lately.  I'm hopeful it will
see a revival at some point as it's a great framework for viewing many
specific dualities, as well as being a fruitful alternative to the more
traditional tools of algebra and coalgebra for representational
applications, see e.g. http://boole.stanford.edu/pub/seqconc.pdf .

Vaughan Pratt

>
> The MBP (aka "the conjugacy principle" which embraces and generalizes
> Jacobi inversion) Ref MR696771
> http://www.ams.org/mathscinet/pdf/696771.pdf  (and no, it does not appear
> in either Wikipedia or PlanetMath, yet) is somewhat heuristic in
> character, suggesting : Transform the problem (T), Solve(S), Transform
> back(T^1), as a "bypass" given by  (T^1)ST, which looks like conjugation.
> Melzak himself refers to adjoints (quite tangentially) as "being bypasses,
> though dressed up and served forth exotically" p.106 ibid. (I do recall
> that adjoints were generally seen as pretty exotic in the early 1970's
> when I was a graduate student at UBC, to my great chagrin).  [...]

Re: exploiting similarities and analogies

[Al Vilcius]

Sorry to be misleading with MBP; perhaps I should have said Melzak's
"Bypass Principle", or Melzak's "Bypass" principle, rather than using the
MBP abbreviation for typing convenience, because both Google and Google
Scholar provide lots of relevant hits to searches without the quotes.
Anyway, the basic reference I have in mind is: Z. A. Melzak, "Bypasses, A
simple approach to complexity" John Wiley & Sons (1983) ISBN 0-471-86854-X

The connection with duality is very appealing, thank you, because
dualities certainly offer some very profound insights. However, I have no
intuition as to why an adjunction suggested by a (Melzak's) Bypass should
have unit and counit as isomorphisms? Should there be a "dualizing object"
lurking about? Are there any known dualities for presheaf categories 
Set^(C^op) where C is something simple like (1 --> 1) or (2 <-- 1 --> 2)
or (1 --> 2 --> 3 <-- 1) etc. (identities omitted). What I'm looking for
is a bypass for gluing presheaves.

Thank you for your kind comments. ...... Al

Al Vilcius
Campbellville, Ontario, Canada






On Tue, April 1, 2008 1:50 am, Vaughan Pratt wrote:
> Al Vilcius wrote:
>> Has anyone explored, either formally or informally, the connection
>> between
>> the Melzak Bypass Principle (MBP) and adjoints?
>
> Considering that Google returned 'Your search - "Melzak Bypass
> Principle" - did not match any documents,' the acronym might be a tad
> premature.  But in the spirit of the question the connection I'd be
> tempted to make (predictably given my biases) would be with duality
> rather than adjunction.  Thus floor: R --> Z as right adjoint to the
> inclusion of Z into R is a (posetal) example of adjunction (with the
> numerical inequalities x <= y as the morphisms) that in this case
> forgets the continuum structure of R, whereas duality being an
> involution (at least up to equivalence) necessarily retains all the
> categorical structure in mirror image.
>
> For example to understand finite distributive lattices, transform them
> to finite posets, work on them in that guise, and transform back at the
> end.  Being a categorical duality, the resulting understanding of
> distributive lattices includes their homomorphisms, which under this
> duality become monotone functions.  Complete atomic Boolean algebras are
> even simpler: viewed in the duality mirror they are just ordinary sets
> transforming by functions.  Hence the homset CABA(2^X,2^Y) of complete
> Boolean homomorphisms from 2^X to 2^Y as complete atomic Boolean
> algebras is (representable as) the set X^Y of functions f:X --> Y  (so
> for example there are 5^3 = 125 Boolean homomorphisms from 2^5 to 2^3).
>   Likewise one can understand Boolean algebras and their homomorphisms
> in terms of their dual Stone spaces and continuous functions, while the
> self-duality of any category of finite-dimensional vector spaces over a
> given field is a linchpin of matrix algebra and essential to the linear
> algebra examples you cited for "MBP."
>
> There is such a diversity of fruitful dualities, of widely varying
> characters, that one despairs of finding any uniformity to them.  For me
> this is where Chu spaces enter.  What I find so appealing about Chu
> spaces is that they tap into a vein of uniformity running through these
> disparate examples whose global structure is that of *-autonomous
> categories, or linear logic when seen "in the light of logic."  (One can
> view *-autonomous categories as being to *-autonomous categories of Chu
> spaces roughly as Boolean algebras are to fields of sets and toposes to
> toposes of presheaves.)  All of the dualities listed above and many more
> can be exhibited as (usually not *-autonomous) subcategories of
> Chu(Set,K) for a suitable set K, with each such category and its dual
> connected by the self-duality of the *-autonomous category Chu(Set,K)
> itself.
>
> http://www.tac.mta.ca/tac/index.html#vol17 , the special issue of TAC on
> Chu spaces that Valeria de Paiva and I edited, conveys some of the
> flavor of this.  Mike Barr gives a history of Chu spaces at
> http://www.tac.mta.ca/tac/volumes/17/1/17-01.pdf, while the preface at
> http://www.tac.mta.ca/tac/volumes/17/pref/17-pref.pdf supplements Mike's
> history and gives an overview of the papers in the volume.  My 1997
> Coimbra notes http://boole.stanford.edu/pub/coimbra.pdf on Chu spaces
> play a more introductory role (the first half anyway, the second half
> emphasized linear logic more than I would have if I were writing it
> today).
>
> The Chu space scene has been a bit quiet lately.  I'm hopeful it will
> see a revival at some point as it's a great framework for viewing many
> specific dualities, as well as being a fruitful alternative to the more
> traditional tools of algebra and coalgebra for representational
> applications, see e.g. http://boole.stanford.edu/pub/seqconc.pdf .
>
> Vaughan Pratt
>
>>
>> The MBP (aka "the conjugacy principle" which embraces and generalizes
>> Jacobi inversion) Ref MR696771
>> http://www.ams.org/mathscinet/pdf/696771.pdf  (and no, it does not
>> appear
>> in either Wikipedia or PlanetMath, yet) is somewhat heuristic in
>> character, suggesting : Transform the problem (T), Solve(S), Transform
>> back(T^1), as a "bypass" given by  (T^1)ST, which looks like
>> conjugation.
>> Melzak himself refers to adjoints (quite tangentially) as "being
>> bypasses,
>> though dressed up and served forth exotically" p.106 ibid. (I do recall
>> that adjoints were generally seen as pretty exotic in the early 1970's
>> when I was a graduate student at UBC, to my great chagrin).  [...]
>
>
>
>
>

exploiting similarities and analogies

[Al Vilcius]

Dear Categorists,

Has anyone explored, either formally or informally, the connection between
the Melzak Bypass Principle (MBP) and adjoints?

The MBP (aka "the conjugacy principle" which embraces and generalizes
Jacobi inversion) Ref MR696771   
http://www.ams.org/mathscinet/pdf/696771.pdf  (and no, it does not appear
in either Wikipedia or PlanetMath, yet) is somewhat heuristic in
character, suggesting : Transform the problem (T), Solve(S), Transform
back(T^1), as a "bypass" given by  (T^1)ST, which looks like conjugation.
Melzak himself refers to adjoints (quite tangentially) as "being bypasses,
though dressed up and served forth exotically" p.106 ibid. (I do recall
that adjoints were generally seen as pretty exotic in the early 1970's
when I was a graduate student at UBC, to my great chagrin). The MBP is
acclaimed in MM vol.57 No.3 May 1984 as "a device for exploring
analogies", or as "a dazzling attempt to comprehend complexity".  Perhaps
"bypass" could also be seen in the words of W.W. Tait (1996) "the
propositions about the abstract objects translate into propositions about
the things from which they are abstracted and, in particular, the truth of
the former is founded on the truth of the latter".
http://home.uchicago.edu/~wwtx/frege.cantor.dedekind.pdf . For me,
"bypass" is a kenning for quasi-inverse or ad joint pairs.

Motivation for seeking such a connection is not really to revive a 25 or
30 year old idea (as brilliant as Melzak's insights were, of course), but
rather "to facilitate invention and discovery" (in Melzak's own words),
and to find additional (as well as interdisciplinary) sources of instances
of adjoints, possibly as a way to make adjoints more immediately relevant
in any introductory discussion of categories, since, of course, adjoints
are undoubtedly one of the most successful concepts within category
theory. Further inspiration could be found in the Brown/Porter "Analogy"
paper http://www.bangor.ac.uk/~mas010/eureka-meth1.pdf and others, along
with a passion for  invention and discovery through the continued pursuit
of Unity and Identity of Opposites (UIO) -  obviously referring to Bill
Lawvere. Furthermore, I would also like to see this connection developed
for practical reasons,  applied to various situations, in particular to
the structure of the www as an anthropomorphic creation that could benefit
from further categorical perspective, given by the learned categorists I
respect the most.

I look forward to your  thoughts and comments. ..... Al

Al Vilcius
Campbellville, ON, Canada