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\title{Restriction categories and categories of partial maps}
\author{Robin Cockett \\ 
        Department of Computer Science \\
        University of Calgary}
\date{1 September 1998}

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\begin{abstract}
Because partial maps are central to so many issues in computer science 
there has been a considerable effort to develop their theory: an early 
categorical attempt was that of Di Paola and Heller who introduced the 
notion of dominical categories.  This was followed quickly by  
Robinson and Rosolini who simplified the categorical structure 
to P-categories, categories with a partial product structure. 
These were also considered by Bart Jacobs as the semantics 
of weakening and Carboni studied them from the bicategorical point of 
view.  

These categories are also, essentially, what I called ``copy 
categories'' in a manuscript which started circulating in about 1995.
It was never published for two main reasons.  First, much of the material 
on partiality was already available in the above (and Mulry's) work.  
Secondly, one of the main motivating results, the description of the 
extensive completion of a distributive category,  had also been 
proven independently by the Steve Lack at roughly the same time.  Thus, 
it had been resolved that we should try to pool resources and publish 
the results jointly .... and that event had to wait for a time when we 
could get together physically!

We did get together in Sydney this last Australian winter and, 
inevitablely, we completely reworked our approach!  Our new approach may be 
regarded as a return to the key ideas expressed in the opening sentence of 
Di Paola and Heller's paper on dominical categories where they emphasized 
the important role of idempotents in expressing partiality.  We are  
now convinced of the value of directly legislating of the existence of 
idempotents to express partiality and we call the resulting algebraic 
structure a restriction category.  

This structure has allowed us to develop surprisingly incisive basic results 
on categories of partial maps in an uncluttered manner.  A slightly  
surprising results is that the category of restriction categories 
${\rm rCat}_0$ is monadic over ${\rm Cat}_0$.  Steve discovered this 
construction while I was trying to show him how formal subobjects (given 
by a fibration) could be universally adjoined to a category.

If there is time I would also like to describe our results on partial 
map classifiers and how these results were greatly facilitated by having 
the theory of restriction categories in hand.
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