What is known about partial morphisms in toposes? A partial morphism from A to B is a morphism from a subobject of A to B. Composition needs certain pullbacks to exist, which they do in a topos, so every topos has a category of partial morphisms. What is known about it? E.g. it isn't a topos, since its initial and terminal objects are the same. Failing that, does it have other nice properties, e.g. limits and colimits? Johnstone's book touches on the subject in chapter 1: all partial morphisms from A to B are representable by (ordinary) morphisms from A to an object B~, and the mapping from B to B~ is a functor with a natural transformation from the identity to that functor. Monads seem about to make an entrance (is there a product transformation from B~~ to B~?), but he doesn't explore the matter any further. Goldblatt's book and Fourman's chapter in the Handbook of Mathematical Logic cover the same ground. What else is known? -- Richard Kennaway SYS, University of East Anglia, Norwich, U.K. Internet: jrk@sys.uea.ac.uk uucp: ...mcsun!ukc!uea-sys!jrk ============================================