From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1945 Path: news.gmane.org!not-for-mail From: jdolan@math.ucr.edu Newsgroups: gmane.science.mathematics.categories Subject: Re: Limits Date: Thu, 3 May 2001 16:38:18 -0700 (PDT) Message-ID: <200105032338.f43NcI203820@math-cl-n03.ucr.edu> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241018223 1336 80.91.229.2 (29 Apr 2009 15:17:03 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:03 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri May 4 09:14:24 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f44BXgl01062 for categories-list; Fri, 4 May 2001 08:33:42 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:1945 Archived-At: |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 may as well state the obvious (not necesarily right) answer to this: no, not quite. rather, what seems to be going on is that the phenomenon of adjoint linear operators is, in yetter's terminology, a sort of decategorification of the phenomenon of adjoint functors. decategorification is generally a somewhat destructive process, destroying the morphisms between objects, and since the morphisms are so intrinsic to the definition of adjoint functor it seems too much to hope for that the decategorified phenomenon of adjoint linear operators could actually qualify as a special case of adjoint functors. there are suggestive indications, though, that all of the really interesting special cases of adjoint linear operators in physics, for example, are decategorifications of interesting pairs of adjoint functors. (for example so-called "creation and annihilation operators on fock space" have categorified analogs that live on a categorified analog of fock space whose objects/vectors are something like joyal's "species of structure".) so roughly: the general phenomenon of adjoint linear operators is technically probably not quite a genuine special case of adjoint functors. the actual interesting special cases of adjoint linear operators, however, are often seen to be mere shadows of more interesting cases of adjoint functors.