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19
Restriction categories I: Categories of partial maps
 Theoretical Computer Science
, 2001
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STABLE MEET SEMILATTICE FIBRATIONS AND FREE RESTRICTION CATEGORIES
"... Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sen ..."
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Abstract. The construction of a free restriction category can be broken into two steps: the construction of a free stable semilattice fibration followed by the construction of a free restriction category for this fibration. Restriction categories produced from such fibrations are “unitary”, in a sense which generalizes that from the theory of inverse semigroups. Characterization theorems for unitary restriction categories are derived. The paper ends with an explicit description of the free restriction category on a directed graph. 1.
Restriction categories III: colimits, partial limits, and extensivity
 Mathematical Structures in Computer Science
, 2007
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Partial Functions, Ordered Categories, Limits and Cartesian Closure
 in: G. Birtwistle (ed), IV Higher Order Workshop
, 1993
"... Introduction Partial maps are naturally ordered according to their extent of definition. Constructions on partial maps should preserve this order so that as a component or module in a construction (such as pairing or composition) becomes more defined then so does the construct as a whole, without c ..."
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Introduction Partial maps are naturally ordered according to their extent of definition. Constructions on partial maps should preserve this order so that as a component or module in a construction (such as pairing or composition) becomes more defined then so does the construct as a whole, without changing any of its existing values. Yet despite the vast literature devoted to partial maps (e.g. [3, 6, 7, 17, 18]) this principle of modularity has not been given systematic attention. To do so the partial maps must be viewed as the morphisms of an ordered category, and the theory of limits, etc. developed in this context. The usual limit theory is inadequate for ordered categories since even if the usual limits exist their constructions will typically violate the order. This can be rectified by the judicious replacement of equations by inequalities (previous approaches to this problem, e.g. lax limits, are inappropriate here). It is expected that these results will
Locally connected recursion categories
, 2006
"... Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of noncomplemented and recursively inseparable domains are given in a locally connected category. The use of local connectedness to produce categorical analogs of undecidable ..."
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Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of noncomplemented and recursively inseparable domains are given in a locally connected category. The use of local connectedness to produce categorical analogs of undecidable problems is new; the approach allows us to relax the hypotheses under which the results were originally proved. The results are generalized to nonlocally
Partial Hyperdoctrines: Categorical Models for Partial Function Logic and Hoare Logic
, 1993
"... In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important ..."
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In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial (first order) hyperdoctrine is a different notion of morphism between the fibres. We also use this partial hyperdoctrine to give a model for Beeson's Partial Function Logic such that (a version of) his axiomatization is complete w.r.t. this model. This shows the usefulness of the notion independent of its intended use as a model for Hoare logic.
RANGE CATEGORIES II: TOWARDS REGULARITY
"... Abstract. In this paper, which is the second part of a study of partial map categories with images, we investigate the interaction between images and various other kinds of categorical structure and properties. In particular, we consider images in the context of partial products, meets and discreten ..."
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Abstract. In this paper, which is the second part of a study of partial map categories with images, we investigate the interaction between images and various other kinds of categorical structure and properties. In particular, we consider images in the context of partial products, meets and discreteness and survey a taxonomy of structures leading towards the partial map categories of regular categories. We also present a term logic for cartesian partial map categories with images and prove a soundness and completeness theorem for this logic. Finally, we exhibit several free constructions relating the different classes of categories under consideration.
RANGE CATEGORIES I: GENERAL THEORY
"... Abstract. Inthis twopartpaper, weundertakeasystematicstudyofabstractpartial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establis ..."
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Abstract. Inthis twopartpaper, weundertakeasystematicstudyofabstractpartial map categories in which every map has both a restriction (domain of definition) and a range (image). In this first part, we explore connections with related structures such as inverse categories and allegories, and establish two representational results. The first of these explains how every range category can be fully and faithfully embedded into a category of partial maps equipped with a suitable factorization system. The second is a generalization of a result from semigroup theory by Boris Schein, and says that every small range category satisfying the additional condition that every map is an epimorphism onto its range can be faithfully embedded into the category of sets and partial functions with the usual notion ofrange. Finally, we give an explicit construction