From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3684 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: dagger? Date: Thu, 8 Mar 2007 23:31:45 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019454 9769 80.91.229.2 (29 Apr 2009 15:37:34 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:34 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Mar 9 09:17:46 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Mar 2007 09:17:46 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HPesJ-0007Dg-Tc for categories-list@mta.ca; Fri, 09 Mar 2007 09:11:12 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 38 Original-Lines: 58 Xref: news.gmane.org gmane.science.mathematics.categories:3684 Archived-At: Lawrence Stout wrote: > > Could someone please post a definition of a dagger category and a > reference for typical useful examples? For reference: an involutive category (or dagger category) is a category C equipped with a contravariant, involutive, identity-on-objects functor "+" (my ASCII rendition of the TeX symbol $\dagger$). The main reason to consider dagger structure is that it allows the following definitions (and other similar ones): * a morphism f:A->B is _unitary_ if f is invertible and f^{-1} = f^+ * a morphism f:A->A is _hermitian_ if f = f^+. * a morphism f:A->A is _hermitian positive_ if there exists some object B and g:A->B such that f = g^+ o g. * a morphism f:A->B is called an _isometry_ if f^+ o f = id. ("Isometry" is to "unitary" like "mono" to "iso"). The main example is the category of finite-dimensional Hilbert spaces. In it, for a map f : A -> B, the map f^+ : B -> A is given as the adjoint of f (in the linear-algebra sense). Note that the definition of the adjoint requires inner products, hence *Hilbert* and not just vector spaces. The category of finite-dimensional Hilbert spaces is additionally compact closed, so that for a morphism f : A -> B, we also have f^* : B^* -> A^*. While the functor (-)^* is also contravariant and involutive, it is not to be confused with the dagger structure. A^* is the dual space, which is not naturally isomorphic to A. Also, relative to chosen bases, the matrix of f^* is the transpose of that of f, whereas the matrix of f^+ is the adjoint (complex conjugate transpose). Dagger compact closed categories were axiomatized by Abramsky and Coecke [LICS 2004], and also by Baez and Dolan [ArXiV:q-alg/9503002, 1995]. One requires the following compatibilities between the two structures: * (f tensor g)^+ = f^+ tensor g^+, * the structural natural isomorphisms (associativity, symmetry, etc) are unitary, * the maps I -> (A^* tensor A) and (A^* tensor A) -> I are each other's adjoints. As Abramsky and Coecke have shown, many interesting constructions from Hilbert spaces can be done in a dagger compact closed category. Another example of a dagger compact closed category is the category of sets and relations, but it is degenerate, in the sense that A^* = A and f^* = f^+. -- Peter