Partial map classifiers and partial cartesian closed categories

Abstract

In this paper we consider two conceptually different categorical approaches to partiality namely partial map classifiers ( pmcs) with total maps, and partial cartesian closed categories ( pcccs) pC with partial maps, showing how these approaches are intimately related. While a topos setting generally provides a richer setting for defining possible pmcs and classes of partial maps, conditions are derived that determine when pmcs and partial maps can in fact restrict to pC. In this way semantic constructs can be interpreted consistently from both approaches. Various examples involving domains and cpos are examined as is the connection to Kleisli categories. Finally, it is observed that a general categorical framework involving monads arises naturally in this context.