Dear Makoto, At 11:08 AM 6/14/2009, you wrote:
I don't know how much Yoneda Lemma is useful in other areas of mathematics, and I have wanted to know it.
The Yoneda Lemma came to mind partly because of M. Barr and C. Wells book "Toposes, Triples and Theories." Its Preface recounts that in the sense of Lawvere's insight -- a mathematical theory corresponds "roughly to the definition of a class of mathematical objects" -- toposes, triples, and theories are beautifully connected fundamental notions. Barr-Wells write that the Yoneda Embeddings Theorem, "the first of several important consequences" of the Yoneda Lemma, "in one way or another is used in practically every mathematical argument in this book." (p. 27) Perhaps subscribers to this list would care to comment on how specific results in this book apply or relate to computer science, other areas of mathematics, logic, or physics. All the best, Ellis D. Cooper [For admin and other information see: http://www.mta.ca/~cat-dist/ ]