Re: exploiting similarities and analogies

[Vaughan Pratt <pratt@cs.stanford.edu>]

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).  [...]