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From: "Prof. Peter Johnstone"
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Subject: Re: "Historical terminology"
Date: Sun, 7 Oct 2007 22:49:22 +0100 (BST)
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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in Peter Freyd's unpublished `pamphlet' "On canonizing category theory;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