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Stream: theory: category theory

Topic: cartesian vs Cartesian

Kenji Maillard (Apr 28 2021 at 12:52):

This is more of a typesetting question rather than categorical one, but what are the guidelines to write the word cartesian (in cartesian categories or cartesian closed structure): with an upper case or a lower case c ? I see that Wikipedia employs the former while nlab uses the latter.

Jules Hedges (Apr 28 2021 at 12:55):

I guess it can only be individual choice. For what it's worth, now you've pointed it out I find Wikipedia's choice extremely weird

Kenji Maillard (Apr 28 2021 at 12:57):

I'm glad to hear that. That was also my gut reaction when my spell-checker tried to correct cartesian to Cartesian, but not being a native english speaker I had some doubts.

John van de Wetering (Apr 28 2021 at 13:19):

Isn't this sort of the highest honour that can be bestowed upon a mathematician: to write your adjectified name with lowercase (second only to perhaps having a lemma named after you). So Abel's commutative groups become abelian groups and Decartes' coordinates system becomes cartesian. So I guess it depends on how much you want to show deference to those gods of mathematics :)

Joe Moeller (Apr 28 2021 at 13:21):

I'm pretty sure Lurie uses "Cartesian" and... "coCartesian" :sick: and so homotopy theorists do too now, and then most other mathematicians use "cartesian"

Mike Shulman (Apr 28 2021 at 13:43):

My experience is that it's category theorists who generally use the lower-case "cartesian", while other mathematicians tend not to.

Mike Shulman (Apr 28 2021 at 13:44):

I do think that once one feels compelled to dualize a person's name, it ought to become lowercase: "coCartesian" is pretty unlovely to my eye. (-:

Joe Moeller (Apr 28 2021 at 13:48):

Mike Shulman said:

My experience is that it's category theorists who generally use the lower-case "cartesian", while other mathematicians tend not to.

Now that you say that, I remember that someone has pointed that out to me before when I was talking about this. I just flipped through an analysis text to see, and it had "Cartesian".

Joe Moeller (Apr 28 2021 at 13:48):

Maybe I keep forgetting this because I don't want it to be true.

Fawzi Hreiki (Apr 28 2021 at 14:17):

Someone once mentioned to me that the greatest honour in mathematics is having a property named after you lowercased, e.g. abelian

Jules Hedges (Apr 28 2021 at 14:18):

So officially this is the "rule". But I don't believe it's really true in practice. In practice, I think you become lowercase by becoming (1) an adjective, and (2) sufficiently common to write

Joe Moeller (Apr 28 2021 at 14:43):

The honor is implicit. People write your name so much they almost forget it's a name, you must have done something right.

Roman Kniazev (Apr 28 2021 at 14:52):

But is there another example, besides "abelian" and "cartesian"? Well, maybe not even from maths.

Joe Moeller (Apr 28 2021 at 14:54):

Apparently "boycott" is from Captain Charles Boycott.

Jules Hedges (Apr 28 2021 at 14:54):

I do it for kleisli

Joe Moeller (Apr 28 2021 at 14:54):

You'd think "Euclidean" would be lowercase, but I don't ever see that.

Joe Moeller (Apr 28 2021 at 14:56):

Relevant: https://english.stackexchange.com/questions/13762/are-there-examples-of-terms-named-after-a-person-that-are-no-longer-capitalized

Nathanael Arkor (Apr 28 2021 at 15:06):

Jules Hedges said:

I do it for kleisli

What about eilenberg–moore?

Roman Kniazev (Apr 28 2021 at 15:07):

Joe Moeller said:

Relevant: https://english.stackexchange.com/questions/13762/are-there-examples-of-terms-named-after-a-person-that-are-no-longer-capitalized

very nice, thanks!

Joachim Kock (Apr 28 2021 at 15:21):

I am definitely in favour of cartesian. Not even Descartes himself wrote his name with a capital C.

John van de Wetering (Apr 28 2021 at 15:26):

Isn't a sandwich named after a person?

Jules Hedges (Apr 28 2021 at 15:27):

They're named after the Earl of Sandwich, who is in turn named after the town of Sandwich, Kent

John van de Wetering (Apr 28 2021 at 15:28):

Do you write Boolean algebra or boolean algebra?

John van de Wetering (Apr 28 2021 at 15:29):

I would definitely write "a boolean" when talking about an element, or the concept in programing languages

John Baez (Apr 28 2021 at 16:41):

Unrelated issue: Johnstone says a "cartesian category" means a category with finite limits, while many other people say it means a category with finite products.

People who want it to mean "finite limits" argue that Descartes introduced not only products like the plane

$\mathbb{R} \times \mathbb{R} = \{(x,y): x,y \in \mathbb{R} \}$

but also more general finite limits, like

$\{(x,y) : x,y \in \mathbb{R}, y = f(x) \}$

I can see both viewpoints, so here is my solution (which I don't expect people to adopt, even though they should):

A category with finite products is a cartesian category.
A category with finite limits is a descartesian category.

Jules Hedges (Apr 28 2021 at 16:49):

I don't know, you unify algebra with geometry after centuries of separation, and people just name all kinds of random stuff after you in the distant future

Nathanael Arkor (Apr 28 2021 at 16:49):

On that same tangent, the people who use "cartesian" for "finitely complete" continue to use "cartesian closed" for "cartesian monoidal closed", which I feel is very awkward.

Jules Hedges (Apr 28 2021 at 16:50):

Another name that's so common that I'm sometimes too lazy to reach for the shift key is frobenius

John Baez (Apr 28 2021 at 16:53):

By the way, Descartes did not invent the "Cartesian coordinates" we know today, where there's an x axis and a y axis:

Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

Jules Hedges (Apr 28 2021 at 16:53):

So more importantly, did Descartes invent the tradition of writing geometry in French?

Jules Hedges (Apr 28 2021 at 16:55):

So I guess we should call categories with finite products "van Schooten categories" then?

Nathanael Arkor (Apr 28 2021 at 16:57):

I'm looking forward to when we can enjoy reading papers about coschootenian categories.

Amar Hadzihasanovic (Apr 28 2021 at 19:18):

I thought I had seen "noetherian" uncapitalised but maybe that was in Italian and French.

Jules Hedges (Apr 28 2021 at 19:19):

Oh yeah, I've definitely seen noetherian, and maybe artinian too

Amar Hadzihasanovic (Apr 28 2021 at 19:20):

(In general in Italian we don't capitalise adjectives -- euclideo, cartesiano, and others are all lowercase)

Nathanael Arkor (May 10 2021 at 15:09):

Another example: hopfian objects.

Martti Karvonen (May 10 2021 at 17:28):

currying as well

Todd Trimble (Jun 04 2021 at 20:10):

Roman Kniazev said:

But is there another example, besides "abelian" and "cartesian"? Well, maybe not even from maths.

There's noetherian.

David Michael Roberts (Jun 05 2021 at 01:40):

I'm in the habit of writing boolean topos, since Boole did nothing with toposes, but would probably write Boolean algebra (Cartesian coordinates, yes, but cartesian categories, by the same logic)