From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3651 Path: news.gmane.org!not-for-mail From: selinger@mathstat.dal.ca (Peter Selinger) Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology: dagger and involution Date: Sat, 3 Mar 2007 01:15:45 -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 1241019434 9634 80.91.229.2 (29 Apr 2009 15:37:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:37:14 +0000 (UTC) To: categories@mta.ca (categories) Original-X-From: rrosebru@mta.ca Sat Mar 3 14:25:59 2007 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 03 Mar 2007 14:25:59 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1HNYlT-0006dQ-SD for categories-list@mta.ca; Sat, 03 Mar 2007 14:15:28 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 108 Xref: news.gmane.org gmane.science.mathematics.categories:3651 Archived-At: Hi Marco and John, thanks for your comments. Although I am not sure how many people this will interest, I should probably try to defend my choice of terminology. I originally invented the term "dagger category" because I was looking for a flexible term that could be used both as an adjective and an adverb. I wanted a term that could be applied not just to categories, but also to many other categorical notions ("dagger categories", "dagger functor", "dagger biproducts", "dagger subobject", "dagger idempotent", "to dagger-split" etc). Abramsky and Coecke had used the term "strongly compact closed category", but "strongly" couldn't be applied in most of these contexts. If I had known about Burgin's erstwhile term "involutive category", I would have probably used it. As it is, I have now been publicly using the term "dagger categories" for over two years, including on this list (first 8 Jun 2005), and the terminology has not drawn any criticism until now (except from John Baez, see below). By now, the term has found its way into published papers, and other have picked it up. So, as John has already pointed out, the proverbial genie has left the bottle. Despite due respect for historical terminology, I have to say that I don't much like the term "involutive category". Most importantly, this leaves no good terminology for categories with an involution that is not identity-on-objects, or not contravariant. I don't much like terminologies that use the name "A" to mean "has properties A, B, and C", just because the first example someone studied happened to have those additional properties. Also, a functor between involutive categories cannot be called an "involutive functor" for obvious reasons. Similarly, one cannot say "involutive idempotent", "involutive biproduct", etc. I think the "dagger" terminology is elegant. As John Baez has pointed out, the term "star category" has ample precedent, and indeed, this shares all the useful grammatical properties of "dagger category". Aside from the fact that star categories are often assumed to satisfy additional properties, the two terminologies are equivalent to each other. The difference comes about because mathematicians write "f^*" for the adjoint of a linear map, whereas physicists write "f^\dagger". So why am I siding with the physicists? The choice was forced by the fact that category theorists have long ago decided to write f^* : B^* -> A^* for the transpose of a linear map f : A -> B (in compact closed categories). This is good notation, because functors should be written the same way on objects as on morphisms. However, this makes it impossible to also write f^* for the adjoint B -> A. So one has no choice but to use f^\dagger : B -> A. The difference between the transpose f^* : B^* -> A^* and the adjoint f^\dagger : B -> A is probably the single most common source of confusion about Hilbert spaces for category theorists and others. Both functors are contravariant, and they have little else in common. Sticking to the term "*-category" would have compounded these problems. Fortunately, the symbol $\dagger$ doesn't already have other meanings in related contexts. So its adoption, at least, should not contradict existing terminology. It is better to have two names for one concept than to have one name for two different concepts. Moreover, since $\dagger$ is only a symbol, and not a dictionary word, there is nothing that prevents it from being pronounced differently by different people. I propose that $\dagger$ can be pronounced (and even translated) as "involutive" by those who prefer to do so. This way, time-honored terminology can be used without a change of notation. -- Peter John Baez wrote: > > On Thu, Mar 01, 2007 at 09:21:55AM +0000, V. Schmitt wrote: > > > John Baez wrote: > > > >by now there's enough literature using the term "dagger-categories" > > >that the genie is out of the bottle. > > > Dear John, just my view: this is not a good argument. > > It's not an argument - I'm just reporting on what I see. > > I don't really like the term "dagger-categories", and I gently > tried to get people to stop using it, but it didn't work. They're > already comfortable with it. > > > I do not know about these dagger categories though > > i read about the compact closed ones. > > So may be I miss the point but, if this is the case, why > > introducing a new terminology if the concepts are not? > > That just creates confusion. > > I hope this is clear: "dagger-categories" are completely different > from "compact closed categories". We need *some* term for them; > we're just arguing about whether to call them "star-categories", > "dagger-categories", or "categories with involution". I like > "star-categories", because in analysis and quantum topology the > special case of "C*-categories" is very important. But, I doubt > we'll reach any sort of agreement! > > Best, > jb > > >