From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1167 Path: news.gmane.org!not-for-mail From: "Robert W. McGrail" Newsgroups: gmane.science.mathematics.categories Subject: RE: Mac Lane's inclusions Date: Fri, 16 Jul 1999 23:11:58 -0500 Organization: Marist College Message-ID: <01BECFE0.BB5BEBA0.robert.mcgrail@marist.edu> Reply-To: "robert.mcgrail@marist.edu" NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241017611 29788 80.91.229.2 (29 Apr 2009 15:06:51 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:06:51 +0000 (UTC) To: "categories@mta.ca" Original-X-From: cat-dist Mon Jul 19 13:12:41 1999 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id LAA11011 for categories-list; Mon, 19 Jul 1999 11:56:34 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Microsoft Internet E-mail/MAPI - 8.0.0.4211 Original-Sender: cat-dist@mta.ca Precedence: bulk Original-Lines: 99 Xref: news.gmane.org gmane.science.mathematics.categories:1167 Archived-At: Colin, Correct me if I am wrong, but it seems to me that every \tau-category is a category with inclusions. Moreover, I recall a result by Freyd that every (sufficiently small? cartesian?) category is equivalent to a \tau-category. The proof does not use choice. In any event, see Categories, Allegories by Freyd and Scedrov for \tau-categories. Hope this helps. Bob McGrail On Thursday, July 15, 1999 12:32 PM, Colin McLarty [SMTP:cxm7@po.cwru.edu] wrote: > This note gives an alternative characterization of "inclusions" in Mac > Lane's sense; and proves every category with selected monics is equivalent > to one with "inclusions". Some of it may have been known before and if so I > would appreciate references. > > In a circular letter Saunders Mac Lane has suggested paying attention to > "inclusions" in categorical set theory. The main questions about this have > general categorical answers as follows. A category has "selected monics" if > for each object C, each equivalence class of monics to C has one selected > representative. The selected monics are called "inclusions" if whenever > C'>->C and C">->C' are selected, the composite C">->C is also selected. > > The following alternative characterization motivates the construction in > the next theorem but is not actually used to prove it. Given monics i and j > to a single object, with i "transition monic" from i to j. (For e-mail convenience I write i i is included in or equal to j.) > > Fact: In any category with selected monics, the selected monics are > inclusions if and only if: for all selected monics i and j with i transition monic h is also selected. > > Proof: First suppose i:I>->C and j:J>->C are selected, and selected monics > compose. The transition h must have some selected equivalent k:K>->J, and so > jk is selected and equivalent to i, which implies jk=i and k=h so h is > selected. Conversely suppose h:I>->J and j:J>->C are selected and transition > monics between selected monics are selected. Then jh:I>->C has some selected > equivalent m and for some iso g we have jhg=m. Thus hg is transition monic > between the selected m and j, and so hg is selected. But hg is also > equivalent to the selected h so hg=h and g is an identity. Thus jh=m is > selected. > > > In any topos with selected pullbacks we have selected monics, since for any > equivalence class of monics we can take the selected pullback of "true" > along the characteristic arrow. These need not be inclusions. However, we > have: > > Theorem: Any category A with selected monics is equivalent to a category AI > with inclusions. > Proof: Let the objects of AI be the selected monics C>->B of A. An AI > arrow from C'>->B' to C>->B is simply an A arrow C'-->C. That is, AI arrows > ignore the monics and just look at the domains. Obviously AI is equivalent > to A and an arrow is monic in AI iff it is monic in A. As the selected > monics to an AI object C>->B we take those AI monics h > > h:C'>-------->C > v v > | | > | | > v v > B = B > > which lie over the same B and make the triangle commute in A. Clearly these > compose, and each monic to C>->B in AI is equivalent to exactly one of these. > > In particular, given any axiomatic theory of a topos with selected > pullbacks: if the axioms are preserved by equivalence, then we can > consistently add the assumption of inclusions. We can assume selected monics > compose. Of course it remains to actually use this assumption to secure the > advantages that Saunders sees for it. > >