From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3386 Path: news.gmane.org!not-for-mail From: F W Lawvere Newsgroups: gmane.science.mathematics.categories Subject: RE: Linear--structure or property? Date: Fri, 11 Aug 2006 10:35:32 -0400 (EDT) Message-ID: References: <039A7CE5BC8F554C81732F2505D511720290F225@BONHAM.AD.UWS.EDU.AU> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019273 8535 80.91.229.2 (29 Apr 2009 15:34:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:33 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Fri Aug 11 14:04:15 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 11 Aug 2006 14:04:15 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GBaMO-0003Om-Ge for categories-list@mta.ca; Fri, 11 Aug 2006 13:59:48 -0300 In-Reply-To: <039A7CE5BC8F554C81732F2505D511720290F225@BONHAM.AD.UWS.EDU.AU> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 118 Xref: news.gmane.org gmane.science.mathematics.categories:3386 Archived-At: Sorry Mike, I believe you misunderstood my definition of "Linear category". It is in my paper "Categories of Space and of Quantity" International Symposium on Structures in Mathematical Theories, San Sebastian (1990), published in the Book "The Space of Mathamtics. Philosophical, Epistemological and Historical Explorations. DeGruyter, Berlin (1992), 14-30. Namely, because "additive" was already established standard usage that included negatives, and because the importance in algebraic geometry etc. of rigs other than rings and distributive lattices had been historically underestimated, I chose the name "linear" because it would at least have an intuitive resonance with physicists and computer scientists and others who routinely apply linear algebra to positive and other contexts. Thus, the definition is simply a category with finite products and coproducts which agree (i.e. there is a zero object which permits the definition of identity matrices, which are required to be invertible). Of course this implies enrichment in additive monoids. However, the question of whether a multiplicative monoid has a unique addition is of interest. I believe it was resolved in the negative by the Tarski school of universal algebra, although I cannot recall the reference, nor whether their examples were as straightforward as Steve Lack's. In ambient categories other than abstract sets, that question has been of interest to me in particular in connection with the foundations of smooth geometry and calculus. Euler's definition of real numbers as ratios of infinitesimals leads immediately to a definition of multiplication as composition of pointed endomorphisms of an infinitesimal object T. This endomorphism monoid R of course has a zero element and hence there are two canonical injections from it into R x R and the requirement on an addition is that it be R-homogeneous and restrict to the identity along both those injections. In classical algebraic geometry, where T is coordinatized as the spectrum of the dual numbers, this addition is unique. In a forthcoming paper I try to approach that result from a conceptual standpoint, by invoking expected functorial properties of integration or differentiation. The matter still needs to be clarified. Concerning Mike's example of the real numbers as a *-autonomous category under addition, I found it useful to note, in my 1983 Minnesota Report on the existence of semi-continuous entropy functions, that the system of extended real numbers, including both positive and negative infinity, is also a closed monoidal category (even though not all objects are reflexive). There are actually two such structures, depending on which sense of the ordering one takes as the arrows of the category; the definition of addition (which is to be the tensor product) must preserve colimits in each variable (where colimits has the two possible meanings). Besides its utility in freely performing certain operations in analysis, this structure strikingly illustrates a point often made to young students: subtraction is just addition of negatives, provided one is in a group like the real numbers; however, in general the binary operation of subtraction can be merely adjoint to addition and, in fact, the condition that A is a "compact" object in a SMC for all B, A* @ B --> hom(A,B)is invertible precisely characterizes the finite real numbers. Best wishes, Bill ------------------------------------------ On Fri, 11 Aug 2006, Stephen Lack wrote: > It's a structure. > > Consider the following category C. > Two objects x and y, with hom-categories > C(x,x)=C(y,y)={0,1} > C(y,x)={0} > C(x,y)=M > with composition defined so that each 1 is an > identity morphism and each 0 a zero morphism, > and with M an arbitrary set. Any commutative > monoid structure on M makes C into a linear category. > > Steve. > > > -----Original Message----- > From: cat-dist@mta.ca on behalf of Michael Barr > Sent: Fri 8/11/2006 6:14 AM > To: Categories list > Subject: categories: Linear--structure or property? > > Bill Lawvere uses "linear" for a category enriched over commutative > semigroups. Obviously, if the category has finite products, this is a > property. What about in the absence of finite products (or sums)? Could > you have two (semi)ring structures on the same set with the same > associative multiplication? > > Robin Houston's startling (to me, anyway) proof that a compact > *-autonomous category with finite products is linear starts by proving > that 0 = 1. Suppose the category has only binary products? Well, I have > an example of one that is not linear: Lawvere's category that is the > ordered set of real numbers has a compact *-autonomous structure. > Tensor is + and internal hom is -. Product is inf and sum is sup, but > there are no initial or terminal objects and the category is not linear. > > > > > > > >