The subject of partial morphisms in a topos has been dealt with extensively in Adam Obtulowicz's Theses which has appeared in Dissertationes Mathematicae. He gives an essentially algebraic definition of what is a category of partial maps in a topos. There is a lot of work on categories of partial maps : Robinson & Rosolini (in Inform. & Comp.) Curien & Obtulowicz (also in Inf. & Comp ) Carboni (Cahiers de Topologie et Geometrie Differentielle) the Theses of Rosolini and Moggi etc. If you are interested not in arbitrary partial maps but in those whose domain of definition is semidecidable you must work with so called dominions. This is a class of subobjects of a topos which contains all iso, is closed under composition and stable under pullpacks along arbitrary maps. The notion of dominion already appears in Rosolini's Thesis. Most prominently it also has been explained by Alex Heller in his paper 'Dominical Categories' which has appeared in J.S.L. Recently Barry Jay has been working on categories of partial maps from the point of view of order enriched categories. I think there are some Edinburgh LFCS repo rts on that. Thomas Streicher ========================