Non-Artin Gluing in Recursion Theory and Lifting in the Abstract Stone Duality for "Category Theory 2000", Como http://hypatia.dcs.qmw.ac.uk/author/TaylorP The analogy between `open' and `recursively enumerable' sets has long been a mainstay of theoretical computer science, but the traditional axiomatisation of general topology involving arbitrary unions is an obstacle to the formal unification of these disciplines. In particular, whilst Artin showed how to glue an open subspace to its closed complement using a comma square of frames, we give a diagonalisation argument to show that this representation fails for any recursively enumerable subset of $\mathbb N$ that is not decidable. Abstract Stone Duality is a re-axiomatisation based on the monadic adjunction between the dual categories of `frames' and `spaces'. Despite the failure of Artin gluing, we show in this formulation that the lift or scone, constructed using a comma square, still provides the classifier for partial maps with open domain of definition. As the axiomatisation is not given in terms of finite intersections and arbitrary unions, a careful study of the modular law is needed to prove this. I would very much like to receive comments and references to relevant work on * Artin gluing and the partial map classifier FOR LOCALES (not toposes), * modular lattices and the (Goursat?) equation ab=bab for idempotents. I will be preparing an abridged version for the Como proceedings during the last week in June, and submitting the paper to "Theory and Applications of Categories" at the same time. The first paper on "Abstract Stone Duality" has been with the TAC referees for nearly a year. Both can be found linked from my Hypatia page. Paul