From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2624 Path: news.gmane.org!not-for-mail From: nobody@nowhere.invalid (Unknown) Newsgroups: gmane.science.mathematics.categories Subject: (unknown) Date: Wed, 29 Apr 2009 15:26:24 +0000 (UTC) Message-ID: <48241.4600825452$1241018785@news.gmane.org> NNTP-Posting-Host: main.gmane.org X-Trace: ger.gmane.org 1241018784 5039 80.91.229.2 (29 Apr 2009 15:26:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:24 +0000 (UTC) Original-X-From: rrosebru@mta.ca Thu Apr 1 14:24:46 2004 -0400 Return-path: Original-Lines: 107 Xref: news.gmane.org gmane.science.mathematics.categories:2624 Archived-At: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Apr 2004 14:24:46 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1B96pj-0006qu-00 for categories-list@mta.ca; Thu, 01 Apr 2004 14:22:31 -0400 From: Todd Wilson MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <16491.20549.171326.902250@localhost.localdomain> Date: Wed, 31 Mar 2004 15:12:05 -0800 To: categories@mta.ca Subject: categories: Re: on the axiom of infinity In-Reply-To: <200403291603.i2TG3UX6000400@saul.cis.upenn.edu> References: <200403291603.i2TG3UX6000400@saul.cis.upenn.edu> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 1 On Mon, 29 Mar 2004, Peter Freyd wrote: > There's a particular operator that keeps popping up for me. > > In an arbitrary heyting algebra define x << y to mean that not only > is x less than or equal to y, but the value of y -> x is as small > as it can be, that is, y -> x = x. In a complete heyting algebra > define an order-preserving, inflationary unary operation s by > > sx = inf{ y | x << y }. > > E.g.: on a linearly ordered set if { y | x < y } has a least element > then that's what sx is. If there is no smallest element above x, > then sx = x (even without the completeness hypothesis). In > particular, note, there no assertion that x << sx. > > The subobject classifier in an elementary topos is complete in the > relevant sense: s is definable. A quick description of the > construction to follow is that we're going to turn s into the > successor operation on an NNO. > > DIVERSION: The definition I just gave is the first I came across. The > next incarnation for me was when I wanted a measure of the failure of > booleaness. In any topos, *A*, there's a largest subterminator B > with the property that the slice category *A*/B is boolean. But > given any subterminator, U, we have its "closed sheaves", *A*_(U), the > full subcat of objects A such that AxU --> U is an iso. (This is a > subcategory of sheaves for a Lawvere-Tierney topology. Starting with > a space X then Sheaves(X)_(U) may be identified with Sheaves(U'), > where U' denotes the complement of U.) Note that the lattice of > subterminators in *A*_(U) is isomorphic to the interval of > subterminators in *A* from U up. We can define BU to be the > largest subterminator in *A*_(U) such that *A*_(U)/BU is boolean. > The interval of subterminators in *A* from U up to BU is boolean > and in the relevant internal sense, BU is the largest such > subterminator. We can, of course, translate this all to a unary > operation on Omega. > > It's the same operator s. > > When one specializes this to a space X it becomes historically > familiar if we dualize it it to a deflationary operator on closed > subsets. It's the operation that removes isolated points. The very > operation that got Cantor started. Hence the word "historically". I haven't yet digested the rest of Freyd's post, but all of the above, including the notation x << y, the connection with collapsing maximal Boolean intervals, the "historical" connection with Cantor, and a lot more, can be found in a series of papers of Harold Simmons: H. Simmons, "The Cantor-Bendixson analysis of a frame", Seminaire de mathematique pure, Rapport no. 92, Institut de Mathematique Pure, Universite Catholique de Louvain, January 1980. H. Simmons, "An algebraic version of Cantor-Bendixson analysis", in Categorial Aspects of Toplogy and Analysis, pp. 310-323, Springer LNM 915, 1982. H. Simmons, "Near-discreteness of modules and spaces as measured by Gabriel and Cantor", J. Pure and Appl. Alg. 56 (1989), 119-162. H. Simmons, "Separating the discrete from the continuous by iterating derivatives", Bull. Soc. Math. Belg. 41 (1989), 417-463. The operation Freyd is calling s (and the associated relation <<) arose in connection with the so-called Reflection Problem for Frames, namely to characterize those frames that have a reflection into the category of complete Boolean algebras. When such reflections exist, they can be found by iterating the functor A |-> N(A), which freely complements the elements of A (and is also the frame of nuclei on A, ordered pointwise), until it "terminates": A -> N(A) -> N^2(A) -> ... -> N^a(A) -> ... (a in ORD). (These maps are all both mono and epi and are components of natural transformations between iterates of N). A basic result here is that N(A) is Boolean iff x << sx for all x in A. The general reflection problem remains open. -- Todd Wilson A smile is not an individual Computer Science Department product; it is a co-product. California State University, Fresno -- Thich Nhat Hanh