From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4016 Path: news.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: Re: connectedness Date: Sun, 14 Oct 2007 19:56:42 -0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019661 11321 80.91.229.2 (29 Apr 2009 15:41:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:01 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Oct 15 10:12:37 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 10:12:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhPiS-0007dd-O8 for categories-list@mta.ca; Mon, 15 Oct 2007 10:10:41 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 73 Original-Lines: 102 Xref: news.gmane.org gmane.science.mathematics.categories:4016 Archived-At: Paul's remarks are quite cogent. Indeed, when Steve Schanuel and I introduced the notion of Extensive category, one of the main motivations was the recognition that a rational theory of connectedness requires a condition on the category of not-necessarily connected things, and moreover that a category like (K-rigs)^op satisfies this=20 condition even though it is not exact. (Also there=20 was the realization of a need for an algebraic=20 geometry for some cases where K is not a ring, indeed where it may satisfy 1+1=3D1). When C^op is an algebraic theory, i.e., C has=20 finite coproducts, then the algebras form a topos iff=20 those coproducts satisfy extensivity (because=20 then the attempt to consider "finite disjoint covers"=20 actually succeeds to satisfy Grothendieck's condition for a"topology"). But a non-trivial dual question is "almost" stated by Paul: For which algebraic categories is the opposite extensive ? Obvious extensions of K-rigs are M-K-rigs, where the=20 given monoid M acts by K-rig homomorphism, and an infinitesimal version of that where M is a Lie algebra acting by derivations. A special case of the question is, given an algebraic category that is coextensive, which varieties in it are also ? (Here I take "variety" in the original Birkhoff spirit, i.e., a full subcategory=20 that is also algebraic for the special reason that it is defined by a quotient theory and is thus closed wrt subalgebras, which a general full reflective algebraic subcategory would not be). A sufficient condition is that the inclusion functor is also COREFLECTIVE. Call these "core varieties". Proposition : A core variety in a coextensive algebraic category is also coextensive in its own right. Hence any core variety is a candidate to serve as the algebras for an algebraic geometry. (Extensivity was the only distinctive feature of rings mentioned in Gaeta's notes on Grothendieck's Buffalo Lectures 1973, and indeed you can verify that the basic=20 construction of a corresponding topos of spaces works, in=20 particular that the algebras become algebras of functions on these). For single-sorted theories, a core variety is defined by the=20 imposition of further identities in one variable having the rare property that the elements satisfying them form a subalgebra. For example ( )^p=3Did in algebras of characteristic p. Already for K=3D2, there are nontrivial core varieties in K-rigs. The best known is the category of distributive lattices, the corresponding topos of spaces being generated by the category of finite posets. The core of any 2-rig is the DL defined by two equations,=20 one of which is idempotence of the multiplication. But the other equation, taken alone, defines a larger core variety whose spaces look like intervals, cubes,etc ; it is intimately related to a less systematic subject burdened with the odd name "tropical". Bill On Fri Oct 12 9:53 , Paul Taylor sent: >Vaughan Pratt's original enquiry was actually in the context of >graph theory (as I suspected at the time, and he subsequently >confirmed), but I would like to add something from the point of >view of constructive real analysis. > >First, though, I would like to underline something that Steve Lack >(almost) said, namely that the category in which you index your >components, and therefore also the one in which you define >connectedness, need to be EXTENSIVE, ie their coproducts should >be disjoint, and stable under pullback, and the initial object strict. > >Maybe we've over-done philology recently, but "component" means >"putting together", where we expect the parts to cover the whole >(coproduct), without overlapping (disjoint), to be distinguishable >(like disjoint union, but unlike addition and disjunction). >The modern notion of extensivity, in which Steve had a part, >captures this idea very neatly. The equivalence between definitions >of connectedness based on 1+1 and on X+Y surely depends on stability >under pullback, and the requirement that the choice between left >and right be unique surely requires disjointness. Maybe a close >study of Marta Bunga's work on abstract connectedness would clarify >this. > >Vaughan originally asked about various categories of algebras, >and Steve mentioned commutative rings, but quietly turned their >arrows around. Stone duality would suggest to me that one should >look for connectedness of algebras in their OPPOSITE category of >"spaces", which I understand in a generic sense that includes >sets, graphs, predomains, locales and affive varieties. > ...