On Sun, 7 Oct 2007, Jean Benabou wrote:
(ii) Your "guess" about cartesian is not correct. Neither in Tohoku, nor in much later papers of his or any of his students, and also by me, was cartesian used in the sense of category with finite limits. If Grothendieck had used this name, which he has not, my "guess" is that he would have called cartesian categories with pull backs , because he and his students used the name "cartesian square" for square which is a pull back. Moreover this is special case of his notion of cartesian map in a fibration.
I first encountered `cartesian' as a synonym for `having finite limits' in Peter Freyd's unpublished `pamphlet' "On canonizing category theory; or, on functorializing model theory" written in about 1975 (I may have got the title wrong, since I no longer possess a copy). However, that paper made it clear that the word was already in use as a synonym for "having finite products"; in it, Peter argued that Descartes should be given credit for having invented equalizers as well as cartesian products. I suspect that its use to mean `having finite products' was a conscious back-formation from `cartesian closed', which undoubtedly dates from Eilenberg--Kelly 1965; but I don't know who first used it in this sense.
(iii) I agree with you on the idea that the "natural" definition of locally cartesian closed category should not imply the existence of a terminal object. If I asked the question, it is because in Johnstone's "Elephant" he does assume a terminal object. Has such an assumption become, now, commonly accepted in the definition ?
I did that because it seemed the appropriate convention to adopt in the context of topos theory. I wasn't trying to dictate to the rest of the world what the convention should be. On the other hand, there seem to be remarkably few `naturally occurring' examples of locally cartesian closed categories which lack terminal objects: the category of spaces (or locales) and local homeomorphisms is almost the only one I can think of. Peter Johnstone