Maps as toposes: terminology?

Maps as toposes: terminology?

[S.J.Vickers]

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
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