Dear All David Leduc’s statement: "A partial functor from C to D is given by a subcategory S of C and a functor from S to D.” If this is taken as given then today I don’t want to add to what people have said. But there is another notion of partial functor that I learnt from Brian Day who learnt it from Bill Lawvere. If we are to replace 2 in Set by Set in Cat, then characteristic maps C —> 2 should become presheaves C^op —> Set, so injective functions S—> C should become discrete fibrations E —> C. Then a partial map from C to D would be a span C <— E —> D with C <— E a discrete fibration and E —> D any functor. The partial map classifier is then the coproduct completion FamD of D. Add size restrictions, to taste. Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]