Re: The Idea of Structure as Data and Conditions

[Michael Barr <barr@math.mcgill.ca>]

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  Has that idea been superceded?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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