Restriction categories as enriched categories

Restriction categories were introduced as a simple equational axiomatisation for categories of partial maps such as those which arise in the foundations of computability theory. A restriction structure on a category is given by an operation obeying four simple axioms, that assigns to each morphism of the category an endomorphism of its domain to be thought of as the partial identity representing the degree of definition.In this paper, we show that restriction categories can be seen as a kind of enriched category; this allows their theory to be studied by way of the enrichment. Unlike most enrichments, ours is based not on a monoidal category, but rather a weak double category in the sense of Grandis–Paré, and provides—from a purely mathematical perspective—an example of such an enrichment arising in nature. Beyond exhibiting restriction categories as enriched categories, we show that varying the base of this enrichment also allows the important notions of join and range restriction category to be understood in the same manner.