Dear John, Sooner or later somebody will call them "sharp" categories, or "tilde" categories... What you are saying is a good argument in favour of a sensible, well established name. Also, on a more general ground, should we have a different terminology in, say: - category theory, - category theory applied to computer science, - category theory applied to physics? Funny names, like quark, can be good and typographical names can be useful, when there is no better substitute. Eg, I do not know of any good substitute for "comma category". But I see no reason to replace a sensible name with a meaningless one; or, even worse, many meaningless ones. --------- Dear Jeff, The problem you are mentioning is essentially based on terminology for different dualities in higher categories. I do not think there is a way of finding a coherent terminology for them, which would not clash with some well established, quite sensible use, already existing in some particular case. Therefore, I would not be surprised if the contravariancy of an involution should assume different meanings in different contexts. --------- All the best Marco