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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

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.

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Abstract. A recursion category is locally connected if connected domains are jointly epimorphic. New proofs of the existence of non-complemented 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 non-locally

this paper we provide a categorical interpretation of the first-order Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the well-known notion of a (first-order) 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. 1. Introduction

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Machine by Curien [Cur86]. It is based upon a weak categorical combinatory logic, viz. lacking surjective pairing and extensionality, that arose as a direct semantic-to-syntactic translation of the lambda calculus of tuples. The computational mode was combinator term reduction through rewriting using a direct left-to-right parse algorithm, initially making the evaluation strategy inefficiently eager 1 . Application is therefore simply juxtaposition, losing the full expressiveness of-reduction that computes via substitution. Its overly strong bias towards the lambda calculus was another factor that limited its expressiveness. On one hand the CAM demanded the existence of categorical products but on the other it had no coproducts for developing many useful data structures. Nevertheless, the high acceptance and efficiency of the CAM-based ML compiler, CAML, gives significant encouragement towards developing a highly-programmable categorical computing paradigm. Some prominent workers in ...

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