From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6572 Path: news.gmane.org!not-for-mail From: Tom Leinster Newsgroups: gmane.science.mathematics.categories Subject: What is Isbell completion? Date: Sun, 6 Mar 2011 22:54:26 +0000 Message-ID: Reply-To: Tom Leinster NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset="US-ASCII" X-Trace: dough.gmane.org 1299519532 25399 80.91.229.12 (7 Mar 2011 17:38:52 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 7 Mar 2011 17:38:52 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Mon Mar 07 18:38:46 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.114]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1PweOM-0001Ve-4B for gsmc-categories@m.gmane.org; Mon, 07 Mar 2011 18:38:46 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:34276) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PweOB-0004qY-NQ; Mon, 07 Mar 2011 13:38:35 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PweO6-00066u-Iq for categories-list@mlist.mta.ca; Mon, 07 Mar 2011 13:38:30 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6572 Archived-At: Dear All, Let A be a small category. Isbell conjugacy gives an adjunction between Set^(A^op) and (Set^A)^op. Like any adjunction, this restricts in a maximal way to an equivalence between full subcategories. Write I(A) for either side of that equivalence. What is I(A)? Certainly I(A) contains the Cauchy-completion of A. But it can be strictly bigger: for example, if A is the initial category then I(A) is the terminal category. Or, if A is a discrete category with more than one object then I(A) is A with initial and terminal objects adjoined. One can also ask the question in an enriched setting. Best wishes, Tom [For admin and other information see: http://www.mta.ca/~cat-dist/ ]