  A restriction category is an abstract formulation for a category of partial
maps, defined in terms of certain specified idempotents called the restriction
idempotents. All categories of partial maps are restriction categories;
conversely, a restriction category is a category of partial maps if and only if
the restriction idempotents split. Restriction categories facilitate reasoning
about partial maps as they have a purely algebraic formulation.
  In this paper we consider colimits and limits in restriction categories. As
the notion of restriction category is not self-dual, we should not expect
colimits and limits in restriction categories to behave in the same manner. The
notion of colimit in the restriction context is quite straightforward, but
limits are more delicate. The suitable notion of limit turns out to be a kind
of lax limit, satisfying certain extra properties.
  Of particular interest is the behaviour of the coproduct both by itself and
with respect to partial products. We explore various conditions under which the
coproducts are ``extensive'' in the sense that the total category (of the
related partial map category) becomes an extensive category. When partial
limits are present, they become ordinary limits in the total category. Thus,
when the coproducts are extensive we obtain as the total category a lextensive
category. This provides, in particular, a description of the extensive
completion of a distributive category.
