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. Section 2 expands on what I wrote about density on this list on June 28 under the heading "Yoneda Theorem < Yoneda Lemma < Dense Yoneda Theorem", giving two characterizations of density that I call respectively semantic and syntactic. (Only two? Kelly gives six characterizations of density in Chapter 5 of his book at http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf .) Section 3 cleans up my several previous attempts, both on this list as far back as 9/7/02 and at CT'04 in Vancouver, at explaining communes, which can be understood as a categorification of Chu spaces as well as a generalization of the Isbell envelope of a category. 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 functors and natural transformations, if possible. 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]