Re: Small is beautiful

[Zinovy Diskin <zdiskin@gmail.com>]

In addition, a key property of semantic categories (as opposed to
syntactic) is their concreteness, i.e., their objects have carriers.
Commutativity with the forgetful functor is essential for results
stating equivalence of syntactic and semantic constructs.

Zinovy

On Tue, Jan 5, 2010 at 12:31 PM, F William Lawvere <wlawvere@buffalo.edu> wrote:
>
> Bob Pare' made the excellent point that not only size but quality
> is relevant.
> I definitely agree with the spirit of his remarks.
>
> Bob happens to have used in passing the term 'syntactic'.
> For clarity, the use of that term needs to be sharpened to avoid
> misunderstanding.
>
> Actually, the term 'syntax' refers NOT to small categories such as
> algebraic theories or rings, but rather to their PRESENTATION by
> signatures or by polynomial generators, et cetera. The process
> of presentation is an adjoint pair quite distinct from the
> semantical adjoint pair: both adjoint pairs have a category of
> theories or of rings in common but are otherwise quite
> independent.
>
> In particular, syntax is NOT the adjoint of semantics. Cratylus,
> Chomsky, and their 21st century followers can be refuted by
> looking soberly at the actual practice of mathematics (wherein
> the construction of sequences of words and of diagrams
> is pursued with great care for the purpose of communication.
> That syntax is only remotely dependent on the structure of the
> content that is to be communicated).
>
> Both of the functors
>
> ?--------------> theories -------------->Large categories
>         Syntax                                      Semantics
>
> are needed.  The domain category of the first can be chosen
> in various useful ways: sketches or diagrams of signatures et cetera.
>
> Happy new year!
>
> Bill
>

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