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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