As is well-known, given a topos Y then the maps f: X -> Y targeted at it are equivalent to toposes in the generalized universe of sets corresponding to Y. Hence there is a correspondence between properties of maps and constructive properties of toposes. Unfortunately, they often have different names. For instance, in "Proper maps of toposes", Moerdijk and Vermeulen have - compact (for toposes) vs. proper (for maps) strongly compact vs. tidy Stone vs. entire compact Hausdorff vs. separated (a.k.a. proper) Furthermore, sometimes f is genuinely to be thought of as a map, with X and Y toposes over some base topos B, and the properties can be relativized to B. For instance, Moerdijk and Vermeulen define "proper relative to B", and then "proper" unqualified means "proper relative to the target (Y)". More names can arise from this; for instance, "proper relative to 1" has been called - at least for locales - "semiproper", "perfect" or even "proper". Does anyone have suggestions of systematic terminology - other than systematically calling everything "proper" - that avoids this proliferation of names? Steve Vickers Department of Pure Maths Faculty of Maths and Computing The Open University ----------- Tel: 01908-653144 Fax: 01908-652140 Web: http://mcs.open.ac.uk/sjv22