Re: "Historical terminology"

Re: "Historical terminology"

[Vaughan Pratt]

JeanBenabou wrote:
> (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..

What most impressed my students and me two decades ago, when we were
applying the concepts of EK65 to modeling concurrency, was their attempt
to define "closed" as a self-contained notion independently of any
tensor product as its left adjoint (or so it seemed to us).  This
defeated us.  Has a clearer story of that attempt, or any related story,
emerged in the meantime?

> (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 ?

Hopefully not.  If affine geometry has no origin, why should locally
cartesian closed categories have a global reference point?  (What would
Andy Pitts have decided there, and for that matter the orientation of
profunctors in B2.7, which seems backwards from say Borceux?)

Vaughan

Re: "Historical terminology"

[Vaughan Pratt]

Vaughan Pratt wrote:
> JeanBenabou wrote:
>> (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..
>
> What most impressed my students and me two decades ago, when we were
> applying the concepts of EK65 to modeling concurrency, was their attempt
> to define "closed" as a self-contained notion independently of any
> tensor product as its left adjoint (or so it seemed to us).  This
> defeated us.  Has a clearer story of that attempt, or any related story,
> emerged in the meantime?

Meanwhile the following examples occurred to me:

1.  Implicational logic without conjunction.

2.  The type structure of the pure lambda calculus without products.

3.  The subcategory of FinSet consisting of the prime powers.

(With regard to 3, Mike Barr mentioned to me that (Eilenberg and?) Kelly
had come up with the category "-6" meaning the category of all sets save
those with six elements, but this seems less natural than the prime
powers, important in ideal theory as we saw in the recent discussion
about the division lattice.)

The free closed category would be a good example if it had ever been
sighted in nature?  Has it?  (Just because we see initial ring every day
in the wild doesn't mean that all free objects arise in nature.)

Vaughan

Re: "Historical terminology"

[Prof. Peter Johnstone]

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

Re: "Historical terminology"

[JeanBenabou]

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
>

"Historical terminology"

[JeanBenabou]

Dear colleagues

I need your help for the following questions:

(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?
(ii) Same questions for "cartesian closed"
(iii) Same questions again for "locally cartesian closed". Moreover,
in this case, does the precise definition imply that such a category
has a terminal object?

Thanks for your help,

Jean
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