From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7899 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Subobject classifier algorithm Date: Wed, 23 Oct 2013 10:52:27 +0100 (BST) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1382538469 32651 80.91.229.3 (23 Oct 2013 14:27:49 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 23 Oct 2013 14:27:49 +0000 (UTC) Cc: Categories mailing list To: Venkata Rayudu Posina Original-X-From: majordomo@mlist.mta.ca Wed Oct 23 16:27:53 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1VYzPd-0001vR-50 for gsmc-categories@m.gmane.org; Wed, 23 Oct 2013 16:27:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:55691) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1VYzOl-000176-T8; Wed, 23 Oct 2013 11:26:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1VYzOk-0007dS-56 for categories-list@mlist.mta.ca; Wed, 23 Oct 2013 11:26:58 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7899 Archived-At: Dear Posina, What you're doing is essentially just using the Yoneda lemma. If your category is a functor category [C,Set], then Yoneda tells you that, for each object c of C, \Omega(c) must correspond bijectively to the set of maps C(c,-) --> \Omega, and hence to the set of subobjects of C(c,-). So you might as well *define* it to be the latter set. [Note that I said `set' rather than `number' as you did, since these sets may very well be infinite.] More generally, if your category is a Grothendieck topos E, then it has a representation as the topos of sheaves on a site (C,J) whose underlying category C can be taken to be the full subcategory of E on a generating set of objects. Then you can similarly conclude that \Omega is (isomorphic to) the sheaf whose value at a member g of the generating set is the set of subobjects of g in E. So the answer to your question `how many objects do we have to check?' is `all the objects in some generating set'. Regards, Peter Johnstone On Sun, 20 Oct 2013, Venkata Rayudu Posina wrote: > Dear All, > > In continuation of the discussion we had sometime ago regarding > algorithms for finding truth value objects, I am wondering if the > following constitutes an algorithm for calculating subobject > classifiers. > > The basic idea is to use the correspondence between parts of an object > and maps to truth value object from the object to find the truth value > object. > > In general we start with an object (of "simplest" shape such as > initial and gradually going to less simple ones), enumerate its parts, > and then look for objects to which the number of maps from the object > is equal to the number of parts of the object. > > In the case of the category of sets, we start with the initial object, > which has one part. Since there is exactly one function from empty > set to every set, this doesn't help in identifying the truth value > set. So we move to [the next] sigleton set, which has two parts. The > set to which there are exactly two maps from the singleton set is a > two-element set, which we take as [candidate] truth value set. > Finally we verify that the two-element set is indeed the truth value > set by way of checking > > parts of an object = maps to truth value object from the object > > in the case of [the set after sigleton set] two-element set. (For now > I'm ignoring the question of how many more objects do we have to > check.) > > The above method does give the correct truth value object in the > categories of maps, graphs, and dynamical systems in addition to the > aforementioned case of the category of sets. > > In the category of [set] maps, we only have to look at two objects > before we get to the terminal object, which lets us identify the truth > value object > > w: D --> C > > where D = {false, u, true} and C = {false, true} > with w(false) = false, w(u) = true, w(true) = true (see Sets for > Mathematics, pp. 114 - 9). > > To give one more illustration, in the case of graphs, we have to go > little beyond terminal object to the generic arrow, whose five parts > correspond to the five graph maps from the generic arrow to the truth > value object of graphs (please see bottom-left corner of the cover of > Conceptual Mathematics). > > In all these case we begin with [the simplest] initial object and go > to next [less simple] object, and at each stage we use > > number of parts of an object = number of maps to truth value object > from the object > > to identify (and then verify) the truth value object. All the more > important is that we have to examine the above correspondence at a few > simple shapes only (beginning with the initial object) to find the > truth value object. > > Would you be kind enough to let me know if there's something wrong in > using the above method to find the truth value object (when there's > one) of a category in general. > > Thanking you, > Yours sincerely, > posina > > http://conceptualmathematics.wordpress.com/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]