Dear Rex, You might be interested in Joyal's arithmetic universes. Paul Taylor's paper http://www.cs.man.ac.uk/~pt/ASD/ASD/insema.pdf and Milli Maietti's papers http://www.math.unipd.it/~maietti/papers/attpub.ps.gz http://www.math.unipd.it/~maietti/papers/attj.dvi are readily available online. They also contain references to other papers. Bas