[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


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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