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). [...]