From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6414 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Terminology of locally small categories without replacement Date: Fri, 10 Dec 2010 10:53:12 +1100 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1291984328 27432 80.91.229.12 (10 Dec 2010 12:32:08 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 10 Dec 2010 12:32:08 +0000 (UTC) To: JeanBenabou , Original-X-From: majordomo@mlist.mta.ca Fri Dec 10 13:32:03 2010 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 1PR28o-0006nQ-5E for gsmc-categories@m.gmane.org; Fri, 10 Dec 2010 13:32:02 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46490) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1PR28Q-0006aM-8C; Fri, 10 Dec 2010 08:31:38 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1PR28N-00069N-IH for categories-list@mlist.mta.ca; Fri, 10 Dec 2010 08:31:35 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6414 Archived-At: Dear Jean Thank you for your message. I have been in Canberra and am only now catching up with emails, reference writing and the like. In the meantime, I see that you provided an answer to the original question under this "Subject" (involving locally small fibrations) and that Richard Garner has responded well to your questions showing the close relationship between the concepts. I think the comma object observation (I would not claim it competitively as "my notion" particularly) was a helpful viewpoint for some researchers. As Richard points out that, to make the definition, the 2-category K does not really need any limits however comma objects are useful (in the same way that internal category can be defined in any category but having pullbacks makes one feel better). (I now notice in my message that C somehow was mistyped later as R. As indicated, more conditions on C give stronger consequences.) I would be pleased to hear what you have in mind as some of the significant and numerous results derivable using locally small fibrations. Walters and I were interested at one time in developing category theory in a 2-category with an analogue P of the presheaf construction ("Yoneda structures"). Mark Weber has recently been able to make use of some of these ideas in developing foundations for recent advances in Batanin's operad theory. I believe my paper [The petit topos of globular sets, J. Pure Appl. Algebra 154 (2000) 299-315] was some help in this respect. In any case, one example of a 2-category K is provided by a finitely complete cartesian closed category E with an internal full subcategory S; we take K = Cat(E) and Pa = [a opposite,S] where [ , ] is cartesian internal hom in Cat(E). In particular, we can take E = Cat so that K is the 2-category Dbl of double categories. Given any internal full subcategory set of Set, there is an internal subcategory fun of Cat which is the double category of squares based on categories in set. The objects of fun are categories in set, the horizontal and vertical morphisms are functors, and the squares are natural transformations in the squares. The small objects of K = Dbl are defined to be the double categories in set. Best wishes, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]