From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3378 Path: news.gmane.org!not-for-mail From: F W Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Laws Date: Tue, 8 Aug 2006 15:19:29 -0400 (EDT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241019269 8502 80.91.229.2 (29 Apr 2009 15:34:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:29 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Aug 9 17:32:36 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 09 Aug 2006 17:32:36 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GAubj-0001wp-29 for categories-list@mta.ca; Wed, 09 Aug 2006 17:24:51 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 81 Xref: news.gmane.org gmane.science.mathematics.categories:3378 Archived-At: A simple answer to Tom Leinster's question involves the Galois connection well-analyzed by Michel Hebert at Whitepoint (2006): in a fixed category an object A can "satisfy" a morphism q: F->Q iff q*: (Q,A) -> (F,A) is a bijection. Then for any class of objects A there is the class of "laws" q satisfied by all of them, and reciprocally. If the category itself is mildly exact, one could instead of morphisms q consider their kernels as reflexive pairs. For example, if there is a free notion, a reflexive pair F' =>F has a coequalizer which could be taken as a law q. However, the "categorical story" that Tom was missing is not told well by the "Universal Algebra" of 75 years ago. Unfortunately, Galois connections in the sense of Ore are not "universal" enough to explicate the related universal phenomena in algebra, algebraic geometry, and functional analysis. The mere order-reversing maps between posets of classes are usually restrictions of adjoint functors between categories, and noting this explicitly gives further information. For example, Birkhoff's theorem does not apply well to the question: "Do groups form a variety of monoids?" Indeed, does the word "variety" mean a kind of category or a kind of inclusion functor? In algebraic geometry, an analogous question concerns whether an algebraic space that is a subspace of another one is closed (i.e. definable by equations) or not. Often instead it is defined by inverting some global functions, giving an open subscheme, not a subvariety, but still a good subspace. The analogy goes still further; a typical open subspace of X is actually a closed subspace of X x R, and of course the category of groups does become a variety if we adjoin an additional operation to the theory of monoids. In my thesis (1963) (now available on-line as a TAC Reprint, and extensively elaborated on by Linton and others in SLNM 80) I isolated an adjoint pair "Structure/Semantics" strictly analogous to the basic "Function algebra/Spectrum) pairs occurring in algebraic geometry and in functional analysis. In that context, note that the epimorphisms in the category of theories (categories with finite products) include both surjections (laws given by equations, dual semantically to Birkhoff subvarieties) as well as localizations (laws given by adjoining inverses to previously given operations, semantically corresponding to "open" algebraic subcategories). Can these "open" inclusions between algebraic categories be characterized semantically? The technical notion "Structure of" was motivated by the example of cohomology operations: in general, the totality of natural operations on the values of a given functor involves both more operations and more laws than those of the codomain category. The example illustrates that such adjoints are of much broader interest than the mere perfect duality that one might obtain by restricting both sides (one does not expect to recover a space from its cohomology, and the category of spaces studied is not even an algebraic category). As an important further example of a large adjoint which specializes both to Galois connections in each space as well as to a perfect duality on suitable subcategories, consider Stone's study of the relation between spaces and real commutative algebras; for computational purposes, the spaces of the form C(X) need to receive morphisms from algebras A (like polynomial algebras) that are not of that form; such homomorphisms are by adjointness equivalent to continuous maps X --> Spec(A), where Spec(A) would map further to R^n if n were a chosen parameterizer for generators of A in a presentation. Best wishes to all. Bill ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************