# Restriction categories and the semantics of partiality

## Next seminars

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

## Summary

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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