Semantics Club 01 06 2001
About five years ago, Martin Hyland and I independently observed a bijective correspondence between Conway (Bekic, or dinatural diagonal) fixpoint operators and traces on categories with finite products. An extra bonus of this correspondence is that the uniformity principle (a la Plotkin) on a fixpoint operator precisely amounts to a uniformity principle on the corresponding trace - see my PhD thesis for the definition and proof. This uniformity principle is general enough to make sense for arbitrary traced monoidal categories. An application of this concept is found in Selinger's work on categorical models of asynchronous communications.
Today I talk about my recent observation that this uniformity principle on traced monoidal categories can be used for constructing new traced monoidal categories (or categories with fixpoint operators) from known ones. The construction is very simple and in some sense old - its origin can be traced back to the Scott induction principle - and seems to enjoy some universal property (as certain limits in an enriched sense; I still need some time to check all the details however). I am still not sure what we can really get from this construction (any suggestion welcome!), but at least this story is good for refreshing our understanding on traces and fixpoints, I hope.