Restriction categories I: categories of partial maps

Abstract

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.

Free Adequate Semigroups

“…Our representation also gives rise to a nondeterministic polynomial time decision algorithm for the word problem in these semigroups; the exact computational complexity of the word problem remains for now unclear, and deserves further study. [3] Free adequate semigroups 367 We show elsewhere [16] that our approach also leads to a description of the free objects in the categories of left and right adequate semigroups (roughly speaking, those semigroups which satisfy the conditions defining adequate semigroups on one side only). An alternative approach to free left and right adequate semigroups appears in recent work of Branco et al [1,11].…”

Algebraic properties of Zappa–Szép products of semigroups and monoids

“…Lawson studied (left) restriction semigroups as the idempotent connected Ehresmann semigroups by drawing connection between semigroup theory and category theory [15]. In the last decade they were studied by Jackson and Stokes [13] in the guise of (left) twisted C-semigroups motivated by consideration of closure operators, and by Manes [16] as guarded semigroups which arose from the restriction categories of Cockett and Lack [4].…”

Many-one reductions and the category of multivalued functions

“…As a means to study partial functions in an abstract setting, p-categories have largely been superseded by restriction categories introduced in [CL02,CL03,CL07]. The primary additional structure of a p-category is its (partial) product, whereas a restriction categories is built upon the mapping of a function to the identity restricted to the functions domain.…”