On 1/14/2012 2:22 PM, Fred E.J. Linton wrote:
Let me refute [that] by channeling the voice of Dana May Latch, the late Alex Heller's student-of-yore, who stumped me once, decades ago, by asking:
: What do you call a category where products and coproducts coincide?
I confessed I had no idea, I had not even an example of that phenomenon,
Fred, you *surely* meant something else by this.
and she immediately offered the example (which I hereby share with George) of sup-complete sup-semilattices (with bottom element (of course)), and sup (and bottom-element) -preserving maps.
Dana was sufficiently taken with CSLat as to publish a short note on some of its properties (JPAA? AU?), prompting Peter Johnstone to write a review of her note to the effect that she should have exploited its self-duality to make her note even shorter. (Peter could have set a good example by making his review a lot shorter. They were both young back then, with all that implies, but come to think of it so were we all, including you, Fred.) One might ask what is the least change to CSLat breaking this property while retaining most of what makes it interesting. I found one answer to this at http://boole.stanford.edu/pub/es.pdf in the course of answering a different question: is there a non-degenerate model of linear logic that models the duality of time and information and of events and states? My suggestion was to leave the (non-empty) meet-join structure of CSLat and its dual unchanged while switching top and bottom. The result was cute and good for a few papers but eventually I saw the light and switched to Chu spaces which answered my original question much better, albeit without as tight a connection to CSLat (it's just a tiny subcategory of Chu(Set,2), and anyway these days I work in Chu(Set,4) when not hacking climate, Euclid, etc.). Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]