From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3992 Path: news.gmane.org!not-for-mail From: "Marta Bunge" Newsgroups: gmane.science.mathematics.categories Subject: RE: What is the right abstract definition of "connected"? Date: Wed, 10 Oct 2007 08:00:15 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed X-Trace: ger.gmane.org 1241019647 11218 80.91.229.2 (29 Apr 2009 15:40:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:40:47 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Oct 10 17:11:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:30 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfhrW-0006au-Oh for categories-list@mta.ca; Wed, 10 Oct 2007 17:08:58 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 49 Original-Lines: 81 Xref: news.gmane.org gmane.science.mathematics.categories:3992 Archived-At: Dear Vaugham, >I'd like to say that "connected" is defined on objects of any category C >having an object 1+1 (coproduct of two final objects). X is connected >just when C(X,1+1) <= 2. > >If this definition appears in print somewhere I can just cite it. If >not is there a better or more standard generally applicable definition I >can use? In Categories and Alligators (1.733, pg 124), an object in a pre-logos is called CONNECTED if it has exactly two complemented subobjects. They observe that in Sh(Y), the terminator is connected iff Y is a connected space. A PRE-LOGOS (pag 98) is a regular category in which Sub(A) is a lattice (not just a semi-lattice) for each A, and in which f#:Sub(B)---> Sub(A) is a lattice homomorphism for each f:A--->B. For a Grothendieck topos e:E---> S (over an arbitrary base S), this definition admits a generalization with "complemented subobject" replaced by "definable subobject", that is, subobjects classified by which, in case S is Boolean, agrees with the Freyd-Scedrov definition. I do not know if this is the sort of abstraction you want. Now for something (not) entirely different: A related notion to the one above is the notion of "abstractly (exclusively) unary" introduced in my thesis (Categories of Set-Valued Functors, University of Pennsylvania, 1966) as part of the definition of an "atom". An object A in a "regular category" X (in the sense of my thesis, which, modulo the stability assumptions is the same as Barr exact) is "abstractly (exclusively) unary" if every A---> \Sum {X_i} in C factors through one (and only one) injection. (The difference with connected is that arbitrary coproducts must be considered and, unlike what I assert in Proposition 11.8, finiteness does not imply this --incorrect use of Zorn's lemma. ) An object A is an "atom" in a "regular category" X if HOM(A,-):X--->Set preserves colimits, thus also the coproducts which exist in X. In particular, A is abstractly (exclusively) unary. More in particular, every A---> B + C factors uniquely trhough one of the injections. The latter is itself equivalent in this context to every A---> 1 + 1 factors uniquely through one of the injections. Just for completeness I state what is shown in my thesis. A "regular" category X is said to be "atomic" is the class of atoms in it is a set and is generating for X. (The funny thing is that almost all the terminology from my thesis was subsequently abandoned -- "atom" was relaced by "A.T.O." (provided exponentiation exists), and "atomic" had a quite different meaning. ) In any case, my theorem reads (all terminology as in my thesis): THM. (Characterization theorem) Let X be any cocomplete atomic regular category. Then there exists a small category C and a functor X--> S^{C^op} which is an equivalence of categories. Conversely every category of set-valued functors S^{C^op} is cocomplete regular atomic. Note: the terminology introduced in my thesis was motivated by the intended theorem which is of the sort "every complete atomic Boolean algebra is isomorphic to a field of sets" (meaning the "field" of all subsets of its set of atoms). There is no published version of my thesis except for microfilms something. The relative version (relative to a monoidal category V) of this characterization theorem is published in Marta Bunge, Relatived Functor Categories and Categories of Algebras, J. Algebra 11 (1), January 1969, 63-101 (communicated by Saunders MacLane). I am sure that I have expanded way more than you would have wanted. Apologies are in order. Cordially, Marta