From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8535 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: Category without objects Date: Sun, 8 Mar 2015 15:53:42 -0400 Message-ID: References: Reply-To: "F. William Lawvere" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1425856500 30603 80.91.229.3 (8 Mar 2015 23:15:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 8 Mar 2015 23:15:00 +0000 (UTC) Cc: categories To: Ronnie Brown Original-X-From: majordomo@mlist.mta.ca Mon Mar 09 00:14:51 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YUkPJ-0003yR-Lj for gsmc-categories@m.gmane.org; Mon, 09 Mar 2015 00:14:49 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58884) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YUkOZ-0005IO-D8; Sun, 08 Mar 2015 20:14:03 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YUkOZ-0004Vz-Dv for categories-list@mlist.mta.ca; Sun, 08 Mar 2015 20:14:03 -0300 Importance: Normal In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8535 Archived-At: It is difficult to understand "without objects" without any definition of "= object". Remember that =2C already before the 21st century=2C modern mathema= tics had begun to overcome medieval metaphysics. In fact =2Cin the late 195= 0s=2C Alexander Grothendieck had made explicit the definition of "subobject= "=2C which seems relevant here=2C as does his powerful legacy of relativiza= tion in several senses. Now we understand that a category C in a category U= is a truncated simplicial object C0->...->C3 satisfying certain limit cond= itions. We are free to call C0 'objects" and C1 "maps" and since C0->C1 is = a subobject of C1=2C we could also say that objects "are" maps=2Cbut "mimic= ked by" seems =10unnecessary (as well as undefined). (Recall that it is actions of such a C in a topos U that form the topos enve= loping=2C as a full subtopos of sheaves=2C the typical U-topos E->U). To give a category "with objects" i=10n a serious sense would seem to be gi= ving MORE than ju=10st a category=2C for example an interpretation as struc= turesC-> B^A=2C the (functor category also emphasized by Grothendieck)of st= ructures of shape A in background B. (Where perhaps B is equipped with an i= nternal embedding in U itself) The case of no structure and featureless background ( which seems to be the= default setting of modern mathematics despite the preference of MacLane's= dear teacher for a vonNeuman-like setting) means in particular that the C0 = in a category there consists of "lauter Einsen" in the sense of Cantor. Those featureless elements X of C0 do obtain a structure by virtue of C1=2C= C2 because taking the latter into account we can see the inside of X as th= e "comma" category C/X involving (not only the subobjects of X and their inc= lusions=2C but also singular figures and reparameterizations) as very exten= sively utilized by Grothendieck . Bill > Date: Sat=2C 7 Mar 2015 14:36:36 +0000 > From: ronnie.profbrown@btinternet.com > To: Uwe.Wolter@ii.uib.no=3B p.l.lumsdaine@gmail.com > CC: categories@mta.ca > Subject: categories: Re: Category without objects >=20 > I remember Henry Whitehead said that he was very impressed by the axioms > for a category in the Eilenberg-Mac Lane paper. >=20 > A curiosity about the definition is that groupoids were defined by > Brandy in 1926=2C and this definition was used by the Chicago school of > algebra and applied to ring theory. Bill Cockcroft told me that the > groupoid notion was an influence. In 1985 I asked Eilenberg about this= =2C > and said no=2C since if it had been=2C they would have used it as an > example! I forgot to ask Mac Lane! >=20 > Ronnie Brown >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]