Peter Freyd wrote:
A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories?
I once answered a similar question. In 1974 or 1975, I published a paper "Adjoint functors induced by adjoint linear transformations" in the Proceedings of the AMS. The idea is that a pair of adjoint linear transformations between two linear topological vectorspaces, e.g., a special case is Hilbert spaces, naturally map to an adjoint pair of functors between the categories which are the lattices of closed subspaces, i.e., a galois connection. Cheers, Paul H. Palmquist Paul Palmquist <Paul_Palmquist@compuserve.com>@compuserve.com> on 05/16/2001 08:35:30 AM To: Paul Palmquist <phpalmquist@west.raytheon.com> cc: Subject: categories: Re: Limits -------------Forwarded Message----------------- From: Dusko Pavlovic, INTERNET:dusko@kestrel.edu To: [unknown], INTERNET:categories@mta.ca Date: 5/10/01 4:29 AM RE: categories: Re: Limits Peter Freyd wrote:
A good question. I have no answer, only a similar (and ancient) question: is there a setting in which adjoint operators on Hilbert spaces can be seen to be examples of adjoint functors between categories?
probably not, but the they seem to be instances of the same general structure. (it is simple, pretty old, and i am sure many have noticed it, but since no one mentioned it, here it goes.) let U : Cat ---> CAT be the embedding of small categories in all, and let Y: Cat^op ---> CAT map each small category A to the presheaves Psh(A). now look at the (pseudo)comma category U/Y. each category A is represented in it by the yoneda embedding A-->Psh(A). the morphisms between A-->Psh(A) and B --> Psh(B) are exactly the pairs of adjoint functors between A and B. on the other hand, let I: Vec---> Vec be the identity functor, and let * : Vec^op ---> Vec take a vector space V to its dual V*. look at the comma category I/*. each hilbert space V is represented in it by the obvious linear map V-->V*. the morphisms between V-->V* and W-->W* are exactly the adjoint pairs of operators between V and W. playing around a bit, these two comma categories can be thought of as Chu(CAT,Set) and Chu(Vec,R) respectively. so both sorts of adjunctions are the instances of the chu morphisms. they are the chu morphisms on the "representation" objects, in the form X --> R^X, where R is the dualizing object. -- dusko PS infact, one could start from Chu(SET,Set), and define categories as the profunctors A-->Set^A which form a monoid with respect to the profunctor composition. you'd get only the object part of the adjoint functors as the morphisms of this chu, but the arrow part follows from the adjunction (i think). now can we characterize hilbert spaces in a similar way within Chu(Vec,R)? this seems to be a completely different kind of question. in particular, it is possible to define "profunctors" with respect to R or C, like we did with respect to Set, and we can compose them, but hilbert spaces do not seem to be monoids with respect to this composition, at least the way it occurs to me. if there is no such composition that they are, then hilbert spaces are like R-enriched graphs, rather than categories. ----------------------- Internet Header -------------------------------- Sender: cat-dist@mta.ca Received: from mailserv.mta.ca (mailserv.mta.ca [138.73.101.5]) by sphmgaae.compuserve.com (8.9.3/8.9.3/SUN-1.9) with ESMTP id HAA02263 for <76600.1050@compuserve.com>; Thu, 10 May 2001 07:29:48 -0400 (EDT) Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f4AAwqm21469 for categories-list; Thu, 10 May 2001 07:58:52 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Message-ID: <3AF9FA78.9968DFEE@kestrel.edu> Date: Wed, 09 May 2001 19:18:32 -0700 From: Dusko Pavlovic <dusko@kestrel.edu> X-Mailer: Mozilla 4.72 [en] (X11; U; SunOS 5.5.1 sun4u) X-Accept-Language: en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Limits References: <200105032338.f43NcI203820@math-cl-n03.ucr.edu> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk