From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6590 Path: news.gmane.org!not-for-mail From: Aleks Kissinger Newsgroups: gmane.science.mathematics.categories Subject: Is this a studied notion of cardinality? Date: Wed, 23 Mar 2011 15:31:55 +0000 Message-ID: Reply-To: Aleks Kissinger NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1300919683 7175 80.91.229.12 (23 Mar 2011 22:34:43 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 23 Mar 2011 22:34:43 +0000 (UTC) To: categories Original-X-From: majordomo@mlist.mta.ca Wed Mar 23 23:34:38 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Q2WdM-0000Kg-El for gsmc-categories@m.gmane.org; Wed, 23 Mar 2011 23:34:32 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:42432) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Q2Wcw-0002aW-RE; Wed, 23 Mar 2011 19:34:06 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Q2Wcr-0001Kn-IM for categories-list@mlist.mta.ca; Wed, 23 Mar 2011 19:34:01 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6590 Archived-At: Let C be a category with a chosen "point" object I (i.e. tensor unit). The "point cardinality" of some object X in C is then the minimum number of points "p : I --> X" required to distinguish any two maps f,g : X --> Y for any Y. Supposing all objects even have a point cardinality implies well-pointedness of the category, but can actually be quite a bit stronger, if in general the point cardinality is much less than | hom(I,X) |. Of course, the thing I have in mind here is dimension of a vector space, where N points are picking out N basis vectors. So, my questions are: 1. is point-cardinality the the most natural generalisation of this notion? 2. does it provide useful information in categories that are bit like vector spaces, like projective spaces or certain kinds of modules of an algebra? Aleks [For admin and other information see: http://www.mta.ca/~cat-dist/ ]