[FOM] Yoneda Lemma as a foundational tool
Vaughan Pratt
pratt at cs.stanford.edu
Sun Aug 9 06:32:52 EDT 2009
The paper "The Yoneda Lemma as a foundational tool" can be downloaded as
http://boole.stanford.edu/pub/yon.pdf
Section 1 attempts to bridge the gap between algebra and category theory
by treating the Yoneda Lemma from the viewpoint of universal algebra.
I'm not sure how interesting this will be however for those favor logic
over algebra as the optimal organization of the foundations of mathematics.
As a natural extension of the Yoneda Lemma, Section 2 gives two
characterizations of density that I call respectively semantic and
syntactic. I propose the latter as having some bearing on the
foundations of mathematics. Again I would expect logicians to be less
likely to find this plausible than algebraists.
Section 3 extends algebras to communes as a kind of algebra consisting
of both elements (as usual) and dual elements (e.g. the open sets of a
topological space, the characters of a group, the functionals of a
vector space, etc.) It also gives some applications of communes to
combinatorics and ontology (shades of categorial grammar!), and
speculates on the origin of the distinction between types and properties.
The "foundational tool" part has to do with my perception of density as
somehow more basic than algebras and homomorphisms. On the theory that
there is little new under the sun that is basic, I would be delighted to
learn that there are even better ways to conceptualize that idea.
Vaughan Pratt
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