Re: Bourbaki and Categories

["Zinovy Diskin" <zdiskin@swen.uwaterloo.ca>]

Dear Michael,

Still some content in Steve's claim could be imagined. A working
mathematician (WM) works with Borubaki's structures like groups or
vector spaces and leaves all worries about what his proofs actually
mean for a working math logician.  For such a WM, sketches (as any
other syntactical machineries) are indeed technical minutiae rather
than mathematical objects. It'd be perhaps a reasonable view unless a
bunch of strong semantic results (Tarski, Mal'cev,Robinson) that our
WM values so much, which are provided by bringing syntax onto the
stage.

Zinovy

On Thu, Sep 18, 2008 at 10:31 AM, Michael Barr <barr@math.mcgill.ca> wrote:
> Of course sketches are mathematical objects in their own right.  Of
> course, the functor that assigns to each sketch the corresponding theory
> is not full or faithful.  But the definition is precise, the notion of
> model is also precise, so I have no idea what, if any, content there is in
> the claim.  Incidentally, you might with equal justice claim that triples
> are not mathematical objects since two distinct triples can have
> isomorphic categories of Eilenberg-Moore algebras.  In fact there are
> triples (or theories) on Set that have infinitary operations, yet whose
> category of models is isomorphic to Set.
>
> Michael
>