From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6068 Path: news.gmane.org!not-for-mail From: John Kennison Newsgroups: gmane.science.mathematics.categories Subject: Re: Question on choosing subobjects consistently Date: Sat, 28 Aug 2010 10:35:18 -0400 Message-ID: References: <201008281224.o7SCO9rh022252@minus.seas.upenn.edu> Reply-To: John Kennison NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1283045124 13878 80.91.229.12 (29 Aug 2010 01:25:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 29 Aug 2010 01:25:24 +0000 (UTC) To: Peter Freyd , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Sun Aug 29 03:25:23 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OpWe9-0008JY-5N for gsmc-categories@m.gmane.org; Sun, 29 Aug 2010 03:25:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39546) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1OpWcl-0006za-DY; Sat, 28 Aug 2010 22:23:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1OpWch-0006dV-U3 for categories-list@mlist.mta.ca; Sat, 28 Aug 2010 22:23:52 -0300 Thread-Topic: categories: Re: Question on choosing subobjects consistently Thread-Index: ActGsupsdpcswzEgRxOyp+8Iz1HUEQAC1KQF In-Reply-To: <201008281224.o7SCO9rh022252@minus.seas.upenn.edu> Accept-Language: en-US Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6068 Archived-At: Peter just said it better, but to clarify what I said: First, it is possibl= e that Grothendieck intended the following: as usual, say that two monos ar= e equivalent if they have the same codomain with an isomorphism between the= ir domains which commutes with the two monos. Choose exactly one mono from= each equivalence class and call it a canonical mono. Then define a subobje= ct as a canonical mono and a subobject of a subobject as the canonical mono= equivalent to the composition. Secondly, it is not always possible to define a class of canonical monos (a= s above) which is closed under composition. Consider the category with 3 ob= jects, A,B,C, in which every hom set. Hom(X,Y) is a subset of the group Z_2= with two elements. Let Hom(A,A) =3DHom(A,B)=3DHom(A,C)=3DHom(B,C)=3DZ_2. L= et Hom(B,A) and Hom(C,B) be empty and let Hom(B,B) and Hom(C,C) each consis= t of just the identity of Z_2. All compositions will be given by the group = operation. Then there are two non-equivalent monos from B to C, so they mus= t both be canonical. The two monos from A to B are equivalent, so only one = of them can be canonical. But the compositions of this canonical mono with = each of the two canonical monos from B to C give us two canonical monos fro= m A to C but they are equivalent.=20 ________________________________________ From: Peter Freyd [pjf@seas.upenn.edu] Sent: Saturday, August 28, 2010 8:24 AM To: categories@mta.ca Subject: categories: Re: Question on choosing subobjects consistently Mike asks how to pick what I'll call "inclusion maps" that is a choice of monics so that each subobject is named by a unique inclusion map and such that the inclusion maps are closed under composition. I don't know what Grothendieck had in mind, but it's an easy application of the stuff about tau-categories in Cats & Alligators copied from my 1975 "pamphlet." The setting there is cats with finite limits but of course you could always work in the full subcategory of representable functors in the category of set-valued contraviariant functors. A tau-structure gives you not only (transitive) canonical subobjects but associative canonical products and all sorts of goodies. But alas it delivers an equivalent category, not necessarily the category you started with. I don't know how to do it in an arbitrary category. But neither does anyone else, not even Grothendieck: Consider the subcategory of the category of sets with three objects, A,B,C each of which is a two-element set. There well be a total of ten maps, three of which, of course, are identity maps. The other seven maps will be one-to-one and onto but only one of them an isomorphism (as defined, of course, in the subcategory). The one non-trivial isomorphism will be on A (necessarily, the twist map). The hom-sets (B,B) and (C,C) each have a single element (necessarily their identity maps). The hom-sets (B,A), (C,A,) and (C,B) are empty. The hom-sets (A,A), (A,B), (A,C) and (B,C) each have two maps, to wit, the two possible one-to-one onto maps. Note that all maps are monic. A has no proper subobjects. B has only one (the two monics from A to B name the same subject of B). C has three proper subobjects one of which is named by each of the two maps from A to C. Each of the other two subobjects of C are named by one of the two maps from B to C (since B has no non-trivial automorphisms they necessarily name different subobjects). Hence, of the four proper subobjects in this category, two of them have just one name, the other two each have two names. No matter how one chooses a choice of "inclusion map" in the latter cases, that is no matter how one chooses a map A -> B and a map A -> C, when you compose the chosen A -> B with each of the two maps from B to C you will get both names for the single proper subobject named by each of the two maps from A to C. They can't both be the chosen map in (A,C). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]