Let me point out that not every structure comes with an obvious notion of morphism. For example, if I just gave the bare-bones definition of topological space, the obvious definition of morphism would be open mappings. On complete lattices, we can have complete homomorphisms, complete sup homomorphisms and, needless to say, complete inf homomorphisms. And I have recently helped characterize the injectives in the category of partially-ordered monoids and marphisms that satisfy f(x)f(y) =< f(xy). There are Heyting algebras. Isomorphisms are always the same, so that is safe. I never understood why the founding paper in category theory was called "The general theory of natural equivalences", when they do consider more general natural transformations. Michael On Fri, 25 May 2012, Ellis D. Cooper wrote:
In the 1952 document at http://mathdoc.emath.fr/archives-bourbaki/PDF/nbt_029.pdf the only mathematician "pr\'{e}sent" referenced by first name only is Sammy.
I was permitted to audit a graduate course on category theory guided by Sammy at Columbia University in the early 1960s. I recall his insistence that mathematical structure is given by data and conditions. Is that idea implicit or explicit in Bourbaki? Has that idea been superceded? How does it relate to the development of algebraic theories as understood by Lawvere, Linton, Barr-Wells, the Elephant, and so on?
Ellis D. Cooper
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