Paul Taylor has given good evidence on why one should avoid using the prefix "weak" when it leads to ambiguities. He says: "A TOPOS is a cartesian category in which each object has a powerobject" (Categories, Allegories, 1.9). Here "cartesian category" is a background definition (genus in philosphical jargon) and "... powerobject" is the distinguishing property we have in mind (species) (cf bounded comprehension) So, if we want to weaken the definition, we do so by weakening the distinguishing property. This gives a different notion of weak topos than McLarty's. In particular, it would have only weak exponentials whereas McLarty's has plain exponentials and a weak subobject-classifier (hence weak powerobjects). La Monte Yarroll says: Why not provide an outside tag to identify the terminology? Colin should use "weak topos" throughout his own work; works by third persons should refer to "weak topos (as per McLarty)". The trouble is, of course, the last phrase will inevitably be changed to "McLarty-weak topos". ==============================================================================