Re: Small is beautiful

[Colin McLarty <colin.mclarty@case.edu>]

I'm not sure I understand this

> 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.

Do you mean that if we choose some kind of sketches for the domain
category then theories are a reflective subcategory, more or less the
'definitionally closed' sketches?   Then a presentation of a theory T
would be (up to isomorphism) any unit arrow of the adjunction with T
as codomain?

best, Colin


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