Information and Computation
Restriction categories I: categories of partial maps
Theoretical Computer Science
Restriction categories II: partial map classification
Theoretical Computer Science - Category theory and computer science
Restriction categories III: colimits, partial limits and extensivity
Mathematical Structures in Computer Science
Boolean and classical restriction categories
Mathematical Structures in Computer Science
Timed Sets, Functional Complexity, and Computability
Electronic Notes in Theoretical Computer Science (ENTCS)
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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-Pare, 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.