Concerning categories of partial maps there are many sources dealing with this subject including [CO], [J], [Mo], [Mu1], [Mu2], [Rom], [Ros], [RR]. The idea that the (~) functor is a strong monad was well known early on in topos theory (check for example Kock and Wraith's notes). In my thesis in 1980 I generalized this idea by constructing refined truth value objects (omega sub p) and their corresponding partial map classifiers (~)p which again were strong monads. These refined truth value objects (and therefore their corresponding subobjects) had to satisfy certain properties including for example closure under composition & pullbacks. Thus m was a p-subobject meant that the characteristic map of m:A p>--> B factored through omega sub p. These constructions were used in several different contexts including describing various refined notions of partial maps, computations and partial higher type objects. Although these were used primarily in a recursive setting, the method was clearly more general. More recently Rosolini, in the general context of p-categories, used the notation dominance to describe such objects [Ros]. This also seems to be a good time to mention the following. One of the most often mentioned categories of partial maps is the partial cartesian closed categories which are supposed to have a terminal object, products and partial function spaces.When I first heard of such categories the partial function spaces seemed strikingly similar to the partial higher type objects defined in my thesis. More precisely the classes of subobjects used to define partial maps are just the p-subobjects for some omega sub p. Of course one is not generally dealing with a topos but there is always one nearby via Yoneda. Thus for any ccc with a refined partial map classifier (~)p the corresponding Kleisli category is equivalent to a pCCC. Conversely every pCCC can be fully embedded inside the Kleisli category of such a category(just take the Kleisli category of the corresponding monad (~)p for the topos). These points and some of the history were found in a preprint I wrote (Partial Map Classifiers and PCCC's) though some details are also mentioned in [Mu2]. Many of the other references also discuss similar ideas though none seem to directly mention the connection to omega sub p and (~)p. [CO] Curien, Obtulowicz, Partiality, Cartesian Closedness and Toposes, Information and Computation, 1989. [J] Jay, Extending Properties to Categories of Partial Maps, LFCS Tech Report 90-107, 1990. [Mo] Moggi, Partial Morphisms in Categories of Effective Objects, Information and Computation, 1988. [Mu1] Mulry, Thesis, 1980. [Mu2] Mulry, Monads and Algebras in the Semantics of Partial Data Types, 1989, (to appear in TCS). [Rom] Roman, On Partial Cartesian Closed Categories, Contemporary Mathematics, vol 92, 1989. [Ros] Rosolini, Thesis, 1986 [RR] Robinson, Rosolini, Categories of Partial Maps, JSL, date ??. ==============================