Re: The Idea of Structure as Data and Conditions

[Charles Wells <charles@abstractmath.org>]

Data and conditions constitute a presentation.  The graph, diagrams, cones
& cocones of a sketch are a presentation.  This idea has not been
superseded, not at all, but it has been completed (in two senses) by the
concept of theory, which is the  object generated by the presentation: The
theory of a sketch, the classifiying topos, the algebraic theory in the
sense of Lawvere, and so on.  This object contains all the information
about any model.

That idea is in some way the other face of, or the complementary point of
view about, data and conditions.

Charles

On Fri, May 25, 2012 at 6:09 PM, Ellis D. Cooper <xtalv1@netropolis.net>wrote:

> In the 1952 document at
> http://mathdoc.emath.fr/**archives-bourbaki/PDF/nbt_029.**pdf<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  Has that idea been superceded?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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