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