- Supervisor
- Prof. Robin Cockett - Department of Computer Science, University of Calgary
- Date and time
- Thursday, December 4, 2008 at 4:30 PM - Caffe' e pasticcini alle 16.15
- Place
- Ca' Vignal - Piramide, Floor 0, Hall Verde
- Programme Director
- Gianluigi Bellin
- External reference
- Publication date
- November 28, 2008
- Department

Partial functions are absolutely fundamental in mathematics. They arises, for example, in elementary calculus. In fact, Karl Menger in his freshman calculus text (Menger 1955) already saw the need for a rigorous abstract theory of partiality. He and his students Bert Schweitzer and Abe Sklar subsequently developed an axiomatic theory for semigroups of partial functions (Menger 1959, Schweitzer and Sklar 1961,1967). Over the next fifty years this axiomatization was reinvented several more times both within semigroup theory and elsewhere. Motivated by the work of Robert Di Paola and Alex Heller (Di Paolo and Heller 1987) , Pino Rosolini and Edmund Robinson (Rosolini and Robinson 1988), and Aurelio Carboni (Carboni 1987), and completely unaware of the volume of work mentioned above Steve Lack and I provided (Cockett and Lack 2002) a categorical axiomatization of partiality which was almost identical to the above. In fact, even the name we chose, {\em restriction\/} categories, had precedents in that literature. However, approaching the subject with categorical tools in hand allowed a much more perspicuous development of the subject than was hitherto possible. In this talk I wish to introduce the basic theory of restriction categories. In particular, I will outline the completeness and representation theorems for these categories and discuss some (unlikely) examples. I will argue that restriction categories provide the cleanest basis, so far, for studying partiality. I will end by discussing some more recent developments. In particular, I would like to introduce Turing categories and the revamping of the theory of computability, which Pieter Hofstra and I are undertaking ... and why it is necessary.

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