Restriction categories I: Categories of partial maps

J. R B Cockett*, Stephen Lack

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

83 Citations (Scopus)


Given a category with a stable system of monics, one can form the corresponding category of partial maps. To each map in this category there is, on the domain of the map, an associated idempotent, which measures the degree of partiality. This structure is captured abstractly by the notion of a restriction category, in which every arrow is required to have such an associated idempotent. Categories with a stable system of monics, functors preserving this structure, and natural transformations which are cartesian with respect to the chosen monics, form a 2-category which we call MCat. The construction of categories of partial maps provides a 2-functor Par:MCat → Cat. We show that Par can be made into an equivalence of 2-categories between MCat and a 2-category of restriction categories. The underlying ordinary functor Par0:MCat0 → Cat0 of the above 2-functor Par turns out to be monadic, and, from this, we deduce the completeness and cocompleteness of the 2-categories of M-categories and of restriction categories. We also consider the problem of how to turn a formal system of subobjects into an actual system of subobjects. A formal system of subobjects is given by a functor into the category sLat of semilattices. This structure gives rise to a restriction category which, via the above equivalence of 2-categories, gives an M-category. This M-category contains the universal realization of the given formal subobjects as actual subobjects.

Original languageEnglish
Pages (from-to)223-259
Number of pages37
JournalTheoretical Computer Science
Issue number1-2
Publication statusPublished - 6 Jan 2002
Externally publishedYes

Fingerprint Dive into the research topics of 'Restriction categories I: Categories of partial maps'. Together they form a unique fingerprint.

Cite this