From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4801 Path: news.gmane.org!not-for-mail From: Paul Taylor Newsgroups: gmane.science.mathematics.categories Subject: Re: terminology in definitions of limits Date: Tue, 20 Jan 2009 16:39:09 +0000 Message-ID: Reply-To: Paul Taylor NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020182 14894 80.91.229.2 (29 Apr 2009 15:49:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:49:42 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Jan 21 09:03:04 2009 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Jan 2009 09:03:04 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LPcgU-00001E-9c for categories-list@mta.ca; Wed, 21 Jan 2009 08:59:54 -0400 Original-Sender: categories@mta.ca Precedence: bulk X-Keywords: X-UID: 22 Original-Lines: 57 Xref: news.gmane.org gmane.science.mathematics.categories:4801 Archived-At: Peter E observed that > each definition of a limit which I've seen contains something > I would describe as a "probe object" or "test object" although I am not sure whether his question is about the name for this (for which either of his suggestions is reasonable), or what. Limits are, of course, examples of right adjoints, and the situation that Peter describes is a case of the adjoint correspondence (considered as a trivial diagram) test object -----> diagram ============================================================== test object ------> limit of diagram So the left adjoint is a "forgetful" functor, which takes the test object and considers it as a trivial diagram, ie with identities as edges. Giving the test object a "name" in the sense of an English word is not such a big deal. However, I would argue that it is important to give it a "name" in the sense of using a particular letter uniformly for it. For this purpose, I propose the Greek letter capital Gamma. The reason for this choice is that the same role is played in symbolic logic by the "context", ie the collection of parameters, along with their types and hypotheses, that occurs in any mathematical statement. In type theory, the letter Gamma is traditionally and uniformly used for this purpose. (Can some type or proof theorist tell me who introduced or established this convention?) Indeed, I use this convention both for this test object and for other parts of the anatomy of an adjunction systematically throughout my book, "Practical Foundations of Mathematics" (CUP, 1999). In so far as there was a previous convention in category theory for the name of this object, it was "U". This came from sheaf theory, where, by the Yoneda lemma, we need only consider maps from hom(-,U), where U belongs to the base category. This category was primordially the lattice of open subsets of a topological space, so the convention came from that of using "U" for an open set. I believe that German-speaking authors were responsible for this, though I don't know what German word it was that began with U. Speaking of sheaf theory, when and to whom was it first apparent that the category of sheaves depends only on the lattice of open sets, and not on the points of a topological space? Paul Taylor www.PaulTaylor.EU pt09 @ PaulTaylor.EU