On Sat, May 26, 2012 at 7:48 PM, Michael Barr <barr@math.mcgill.ca> wrote:
Let me point out that not every structure comes with an obvious notion of morphism.
This is entirely true on Bourbaki's theory of structures. Mike gives topological spaces as a good example. Should open sets be preserved by morphisms? or reflected? or both? Each Bourbaki structure comes with a unique obvious notion of isomorphism: a bijection which preserves and reflects all the data. Each one comes with as many possible definitions of morphism as there are ways to choose which data to preserve and which to reflect. Weil correctly understood this, as shown in the quote: \begin{quotation} As you know, my honorable colleague Mac~Lane maintains every notion of structure necessarily brings with it a notion of homomorphism, which consists of indicating, for each of the data that make up the structure, which ones behave covariantly and which contravariantly [\dots] what do you think we can gain from this kind of consideration? (Weil letter to Chevalley 1951).\end{quotation} But Saunders was not thinking of any formal definition of "structure". He meant that in fact wherever you see mathematicians using some notion of space or algebra, or whatever, you will see a notion of homomorphism used with it. And he was largely right, though he also helped to make this more strictly true by convincing people it was a useful perspective. There was by then a well-established notion of topological space with continuous functions as morphisms (reflecting open sets). An important subclass of morphisms was open functions, defined as continuous and preserving open sets. Nobody studied topological spaces plus functions which merely preserve open sets (without also reflecting them). And Saunders knew that homomorphisms in his sense need not be functions in the strict sense. He knew algebraic geometry used functions which are not everywhere defined. Functional analysis used "functions" which have no well defined value at any single point. Weil correctly understood that a structure in his sense admits many different notions of morphism. But he was committed to a general set theoretic theory of "structure," so he struggled with that, and got Bourbaki to struggle with it (as seen in Le Tribu in 1952), and Bourbaki failed to make anything useful of it. The more categorical members of Bourbaki never achieved a general theory of "structure" either. But they created many extremely useful and now widely used categorical tools. I will mention that recent philosophers of mathematics have been satisfied with simpler set theoretic theories of structure where data are always preserved. These philosophers are little aware of topological spaces, let alone say uniform spaces or local rings, and so unaware of the complexities Bourbaki faced. colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]