Cher Fred, Merci pour ta reponse rapide. Although your french is perfect, I shall continue in english, for the persons who are less familiar with french. (i) Your "guess" about cartesian closed categories is most certainly correct. I knew that Eilenberg/Kelly had explicitly used this name in their La Jolla paper, and it is probably the first instance, because "closed", in this sense, was first introduced in that paper, as far as I know.. (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. (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 ? Thanks again, to you of course, and to whoever will help me to clarify (ii) and (iii) Jean
Salut, Jean,
Without references at hand to consult, other than my failing memory, I venture to hazard the following GUESSES at answers:
(i) Who gave the name of "cartesian" to categories with finite limits? When was this name given? What is the first published paper where this name occurs?
This name I thought either you, or perhaps earlier Grothendieck, had coined. When? Where? no idea (but if Grothendieck, then Tohoku?).
(ii) Same questions for "cartesian closed"
My unverified guess: Eilenberg/Kelly, La Jolla, 1965.
(iii) Same questions again for "locally cartesian closed".
No idea, but rather much later.
... Moreover, in this case, does the precise definition imply that such a category has a terminal object?
Here I have no answer at all, sorry, beyond this: IF the definition of LCC is just that each "slice" category (but not necessarily the category itself) be cartesian closed, then most probably NOT.
Thanks for your help,
I can only hope you find my guesses WERE actually of any help. I fear, though, that they probably weren't at all. I'd be very interested in learning the outcome of your survey, however.
Jean
Cheers,
-- Fred