From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6593 Path: news.gmane.org!not-for-mail From: soloviev@irit.fr Newsgroups: gmane.science.mathematics.categories Subject: Re: Is this a studied notion of cardinality? Date: Fri, 25 Mar 2011 14:33:41 +0100 (CET) Message-ID: References: Reply-To: soloviev@irit.fr NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1301140017 16351 80.91.229.12 (26 Mar 2011 11:46:57 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 26 Mar 2011 11:46:57 +0000 (UTC) Cc: "categories" To: "Aleks Kissinger" Original-X-From: majordomo@mlist.mta.ca Sat Mar 26 12:46:48 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Q3Rx8-0000ei-NN for gsmc-categories@m.gmane.org; Sat, 26 Mar 2011 12:46:46 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:46401) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Q3Rwh-0005Sd-JB; Sat, 26 Mar 2011 08:46:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Q3Rwe-0004uE-R4 for categories-list@mlist.mta.ca; Sat, 26 Mar 2011 08:46:16 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6593 Archived-At: Dear Alex, I had some papers where the fact that tensor unit I is a generator (i.e., your "point cardinality" is 1) is used to describe ALL natural transformations between superpositions of distinguished functors (for example, tensor and internal hom in symmetric monoidal closed categories, or in compact closed categories) S.V. Soloviev. On natural transformatioms of distinguished functors and their superpositions in certain closed categories.-J.of Pure and Applied Algebra 47(1987) p.181-204. or of superpositions of tensor and biproduct Robin Cockett, Martin Hyland, Sergei Soloviev. Natural transformation between tensor powers in the presence of direct sums. Rapport de recherche, 01-12-R, IRIT, Universit=C2=B4e Paul Sabatier, Toulouse, juill= et 2001. The technique can be used in case of "multiple generators" (your "point cardinality" > 1) but was never detailed as a paper. This is about possible applications of the notion you suggest. Regards Sergei Soloviev > 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/ ]