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Stream: practice: terminology & notation

Topic: Morphisms for which all cartesian lifts exist

Max New (Nov 03 2025 at 16:05):

Let $F : C \to D$ be a functor (not assumed to be a fibration). Is there a common term for a morphism $f : x \to y$ in $D$ such that for all $F(\hat y) = y$ the cartesian lift $f^*\hat y$ exists?

Nathan Corbyn (Nov 03 2025 at 17:04):

I don’t know of one in general, but in case $F = \mathrm{cod} : \mathcal{D}^\to \to \mathcal{D}$ we would say $f$ is carrable. So maybe $F$-carrable?

Max New (Nov 03 2025 at 17:40):

carrable seems fine. Any explanation for why it's called that?

Nathanael Arkor (Nov 03 2025 at 17:58):

It's a synonym of "quarrable" in French, which means "quadrable", presumably because if a morphism is such, then we can form universal squares?

Mike Shulman (Nov 08 2025 at 18:50):

When writing in English, I think we should generally translate words into English. So maybe "$F$-quadrable"?

John Baez (Nov 09 2025 at 14:22):

Of course "quadrable" is a half-English, half-Latin version of the word "squarable". So you could use "squarable" too, and it would be even clearer that it has something to do with squares. (When I see "quadrable" it reminds of expressions like "solvable with quadratures", which means "integrable".)

But I don't really care - it's just another possibility.

Kevin Carlson (Nov 10 2025 at 18:59):

There is a pretty substantial history of papers using "carrable" in English, but it does seem to mainly trace back to Francophone writers.

Kevin Carlson (Nov 10 2025 at 19:00):

Interesting tidbit: I didn't realize till now that "square"'s etymology actually goes back to "quadra"! On reflection it doesn't look very germanic, but it's a bit sneaky to have dropped the "d".

John Baez (Nov 10 2025 at 22:00):

I once thought I saw some evidence that "Descartes" is a shortening of "Des Quatres", thus explaining why he liked graph paper so much. But now I'm seeing a different story.

Mike Shulman (Nov 11 2025 at 07:31):

John Baez said:

Of course "quadrable" is a half-English, half-Latin version of the word "squarable".

True, but at least the Latin prefix "quadr-" is pretty well integrated into English already (quadratic, quadrilateral, quadruple, quadricep, quadrangle, quadricycle, ...). I don't think the same can be said for "carr-".

John Baez (Nov 11 2025 at 17:31):

Definitely. Quadrable is to squarable as regal is to kingly. Kinda.

Kevin Carlson (Nov 14 2025 at 19:05):

No, "quadrable" is to "squarable" as "regal" is to "royal", I think. The analogue of "kingly" would be something like "four-sides-able", which for some reason hasn't caught on.