From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5810 Path: news.gmane.org!not-for-mail From: Jeff Egger Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax_monoidal_functors Date: Sat, 15 May 2010 09:23:14 -0700 (PDT) Message-ID: Reply-To: Jeff Egger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1274023954 18889 80.91.229.12 (16 May 2010 15:32:34 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 16 May 2010 15:32:34 +0000 (UTC) To: =?iso-8859-1?Q?_Andr=E9Joyal?= , Michael Shulman Original-X-From: categories@mta.ca Sun May 16 17:32:33 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ODfpP-0008Rm-E7 for gsmc-categories@m.gmane.org; Sun, 16 May 2010 17:32:31 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ODfLZ-0006Xy-7e for categories-list@mta.ca; Sun, 16 May 2010 12:01:41 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5810 Archived-At: >> I guess that in the category of R-modules over a=0A> > commutative ring= R,=0A> > a module M has a (good) dual iff it is finitely=0A> > generated p= rojective=0A> > iff the endo-functor functor Hom(M,-) preserves all=0A> > c= olimits=0A> > (M is *compact* in a strong sense).=0A=0AObviously this is co= rrect. But, on the other hand, Rel is a =0Acompact closed category (also: = V-Prof, for suitable choice =0Aof V). So it is not necessarily the case th= at every object =0Aof a compact closed category is small/finite/compact. = =0A=0A> Indeed, but in this case it is the objects of the category=0A> whic= h are=0A> "compact," not the category itself.=A0 So if this is the=0A> argu= ment, then=0A> a more natural term would be "locally compact" (clashing=0A>= with "locally=0A> small," of course, but agreeing with "locally presentabl= e"=0A> categories=0A> in which all objects are presentable).=0A=0AHmmm, eve= n that last point is pretty tenuous... A locally =0Apresentable category m= ay have the property that every object =0Ais presentable, but the converse = is false. For example, Sup =0A(the category of complete lattices and supre= mum-preserving =0Amaps) is not locally presentable; but it is monadic over = Set=0Aand therefore has the property in question. =0A=0A> (I am *not* propo= sing to *actually* use "locally compact"=0A> -- I don't=0A> want to introdu= ce yet another name for something that=0A> already has at=0A> least four na= mes, even if none of the existing four are=0A> optimal.)=0A=0AI disagree wi= th this line of argument: if good terminology=0Acan be found, it will kill = off its rivals PDQ. In fact, I =0Ahave not been able to stop myself from t= hinking about this=0Aissue, and would like to propose "simply closed catego= ry" as =0Aa replacement for "autonomous category" (in the sense of: =0Amono= idal category in which every object has a left and a =0Aright dual). The p= oint is that such a monoidal category is =0A(both left and right) closed; m= oreover, it is one in which =0Athe "closed structure" (i.e. the pair of int= ernal homs) =0Aadmits an unusually simple description. =0A=0AOne possible = objection, aside from that which Mike has =0Aalready made, is that the word= "simple" already has an =0Aestablished mathematical meaning. My rebuttal = to this is =0Athat there are precedents for using an adverb independently = =0Aof the corresponding adjective. For example, I see no =0Aconnection bet= ween the "completely" in "completely positive =0Amap" and any of the standa= rd meanings of "complete". =0A=0ACheers,=0AJeff.=0A=0A=0A [For admin and other information see: http://www.mta.ca/~cat-dist/ ]