From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3957 Path: news.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: Re: Ideal Theory 101 [was: is 0 prime?] Date: Fri, 05 Oct 2007 13:59:00 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019626 11083 80.91.229.2 (29 Apr 2009 15:40:26 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:26 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLaP-0003I0-CF for categories-list@mta.ca; Sat, 06 Oct 2007 23:09:41 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 14 Original-Lines: 139 Xref: news.gmane.org gmane.science.mathematics.categories:3957 Archived-At: As I emphasized under (2) in my Sep29 posting,=20 the point of view or perspective on the Kummer=20 functor that factors it through the large category=20 of module categories is quite interesting and useful and thoroughly understood by categorists, and so hides no "mysteries" of a general nature. Jeff has=20 reiterated that point now in elegant detail.=20 But my point was that another perspective, at least=20 as important and at least as old, is perhaps not=20 yet so well explained categorically. Categories of spaces=20 are often analyzed in terms of algebras of functions,=20 hence subspaces in terms of epimorphisms of algebras, (localizations for open subspaces and) regular=20 epimorphisms for closed subspaces. Of course the corresponding congruence relations can sometimes be identified with ideals in some sense. But algebras may be something different from commutative rings, in particular there may be no (known) categories of "modules" in which they can be=20 identified with monoids (an important example is=20 Cinfinity spaces and algebras). Yet the concrete example of the category of commutative rings should give clues toward understanding the geometric phenomenon that another operation besides the lattice ones crops=20 up naturally on the closed subspaces in all these categories. The understanding sought is thus primarily about these=20 categories themselves. (The example contrasting a relief map with a flat paper one was mentioned to show that these infinitesimals are real.) "All these categories " includes many algebraic categories, but not all. For example in the simplest algebraic category (no operations), the only "resolution" of the contradiction between intersection and image is surely trivial ? On Fri Oct 5 12:10 , Jeff Egger sent: >I thought that I had explained my point of view clearly=20 >enough, but apparently I haven't.=20=20 > >If A and B are (additive) subgroups of a ring R (commutative >or otherwise), then A.B is the image of the composite=20 > A @ B ---> R @ R --m-> R >(where @ denotes tensor product of abelian groups, and=20 >m is the multiplication of the R, regarded as an arrow=20 >in AbGp). What is mysterious about this?=20=20 > >We have a functor AbGp ---> Pos which maps an abelian group >X to its set of subgroups; this uses only the existence of=20 >an appropriate factorisation system on AbGp. It is, in fact,=20 >also a monoidal functor with Sub(X) x Sub(Y) ---> Sub(X @ Y)=20 >defined by (A,B) |-> (the image of) A @ B ---> X @ Y. > >Now regarding a ring as a monoidal functor 1 ---> AbGp,=20 >we obtain a composite monoidal functor 1 ---> Pos, which=20 >is a monoidal poset. Specifically, the multiplication=20 >on Sub(R) is defined by=20 > Sub(R) x Sub(R) ---> Sub(R @ R) --Sub(m)--> Sub(R) >which is exactly what I described earlier.=20=20 > >The mystery, if there is one, is why this monoidal poset >happens to be closed. My explanation is that the monoidal >functor AbGp ---> Pos factors (as a monoidal functor) through >the monoidal forgetful functor Sup ---> Pos. This can be=20 >easily derived from the fact that AbGp is cocomplete and @=20 >cocontinuous in each variable; in fact, weaker hypotheses=20 >would seem to suffice.=20=20 > >Thus, again regarding a ring as a monoidal functor 1--->AbGp, >we can consider the composite monoidal functor 1--->Sup; which=20 >is nothing more nor less than a monoidal closed poset that=20 >happens to be (co)complete---and its underlying monoidal poset=20 >(i.e., the composite 1--->Sup--->Pos) is Sub(R) by definition=20 >... mystery solved! > >Not quite, I hear you say: we want ideals, not arbitrary=20 >additive subgroups---but Sub(R) retains enough information=20 >of R to remember which of its elements are ideals and which=20 >are not: an ideal is an additive subgroup A such that=20 >T.A=3DT=3DA.T, where T denotes the top element of Sub(R)---namely, >R itself. This is the "second stage" referred to in my=20 >"Quantale Theory 101" post, which is the process of turning >a quantale into an "affine" quantale (one whose top element=20 >is also its (multiplicative) unit). In the commutative case, >at least, this poses no problem whatsoever. > >I hope this clarifies my point of view on the Kummer functor. >I don't see its existence as having anything in particular to=20 >do with commutative rings, but rather with those properties of=20 >AbGp which cause the monoidal functor AbGp--->Pos to=20 > a) exist, and > b) factor through the monoidal forgetful functor Sup--->Pos >---which, as I sketched above, are not particularly rare ones. >The rest is taken care of by a purely quantale-theoretic process. >[But my comment about Sup being an awesome category had more to do >with why the Kummer functor "should be" interesting (aside from=20 >the obvious concrete considerations), rather than why it exists.] > >I admit that this isn't entirely satisfying if you really are=20 >interested in ideals as representing quotients of a ring; but=20 >I do think that it is a valid perspective, nevertheless, and=20 >welcome further discussion on the topic. > >Cheers, >Jeff. > >--- wlawvere@buffalo.edu wrote: > >> The awesome nature of Sup cannot be the reason why >> the Kummer functor exists, since it is merely used for=20 >> recording the result. The functor is "caused" rather=20 >> by an internal feature of the domain category C of=20 >> commutative rings: The category of quotient objects >> of any given R has a binary operation * that is neither >> sup nor inf even though in principle it can be=20 >> expressed as a combination of limits and colimits. >> We can call it R/ab=3DR/a *R/b but how does the=20 >> operation * specialize to C concretely ? >>=20 >> Bill > > > > Get a sneak peak at messages with a handy reading pane with All new = Yahoo! Mail: http://mrd.mail.yahoo.com/try_beta\?.intl=3Dca > >