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 ============================================
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 ========================
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Thomas Streicher