I find this paper very helpful as someone trying to understand the applications of density: the idea that density gives rise to algebraic analogues of presheaves is appealing. For example I have been unable to get directly a nice dense subcategory of the category of crossed complexes, but have verified that for the equivalent category of cubical omega-groupoids with connections the full subcategory on the free such gadgets on n-cubes is dense. Thus dense sub categories seem a basic tool for `categories for the working mathematician'! A paper is in preparation on density and crossed complexes with an application to tensor products, with Ross Street. On the more philosophical points about types and properties, I tend to look at this from my experience (not wholly good, but way back) with the computer algebra system AXIOM. The notion of `type' was not defined clearly, though numerous examples were in the system, but `type' included a signature (is this a `property'?) , which specified the `types' of the outputs, the `types' of the inputs, and the whole `type' included information on axioms, records, algorithms, printing, etc. I have often wondered what is the proper theory of these `types', as used in AXIOM. I like also the point about the relevance of evolution. The geneticist Dobzhansky wrote a famous paper with title `Nothing in biology makes sense except in the light of evolution', (American Biolology Teacher, 1973) and I think we have to accept that mathematics is a biological product (from humans, I suppose). I have said a little about this in the second of two presentations downloadable from www.banngor.ac.uk/r.brown/askloster.html which points out also a report of the amazing computational properties of the cricket nervous system. Ronnie ________________________________ From: Vaughan Pratt <pratt@cs.stanford.edu> To: categories list <categories@mta.ca> Sent: Sunday, 9 August, 2009 11:21:04 AM Subject: categories: Yoneda and density 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/ ]