From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5160 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: question Date: Tue, 22 Sep 2009 12:14:59 +1000 Message-ID: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1253622308 16308 80.91.229.12 (22 Sep 2009 12:25:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 22 Sep 2009 12:25:08 +0000 (UTC) To: jim stasheff , Categories list Original-X-From: categories@mta.ca Tue Sep 22 14:25:01 2009 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 4.50) id 1Mq4QW-0003q4-46 for gsmc-categories@m.gmane.org; Tue, 22 Sep 2009 14:25:00 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Mq3xN-0002er-5Y for categories-list@mta.ca; Tue, 22 Sep 2009 08:54:53 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5160 Archived-At: Jim: I don't understand your context precisely. (I heard a talk the other =20 day where the speaker used "category" to mean "A_{infinity}-category" =20= without any explanation.) However I can tell a story which uses some =20 of the words you have. Without the axiom of choice (such as in a topos), there are two =20 different conditions on a functor f : A --> X for it to be an =20 "equivalence": 1) there is a functor g : X --> A such that f g and g f are isomorphic =20= to identity functors; and, 2) f is full, faithful and essentially surjective on objects (this =20 last means each object of X is isomorphic to a value of f). Clearly 1) implies 2). The converse holds when epis split (Ax Choice) =20= in the ambient world. Stacks are designed not to see the difference between equivalences of =20= types 1) and 2); that is, if you hom out of an equivalence of type 2) =20= into a stack [for an appropriate topology], you get an equivalence of =20= type 1). See old papers of Par=E9, Bunge, Joyal, . . . Ross On 20/09/2009, at 11:21 PM, jim stasheff wrote: > What do you call it when you have one (small) category being a =20 > (full) subcategory of another , and every object in the big category =20= > is isomorphic to one in the small category ? This is the case for =20 > the category given by objects hom(S,A) ,and morphisms given by the =20 > equivalence > relation hom(T,A) ,as a subcategory of stack(A) . Is there an =20 > equivalence of categories ? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]