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Stream: learning: questions

Topic: "contravariant" 2-functors

Nathanael Arkor (Dec 01 2020 at 20:16):

A functor $\mathbf C^{\mathrm{op}} \to \mathbf D$ is called a contravariant functor from $\mathbf C \to \mathbf D$. Is there an analogous name for a 2-functor $\mathscr C^{\mathrm{co}} \to \mathscr D$?

John Baez (Dec 01 2020 at 20:19):

An opvariant functor, obviously. :laughing:

Oscar Cunningham (Dec 01 2020 at 20:22):

Shouldn't it be an optravariant functor?

Nathanael Arkor (Dec 01 2020 at 20:23):

Whoever decided on the term "covariant" was not thinking ahead.

John Baez (Dec 01 2020 at 20:25):

Back then, probably in the late 1800s, the "co-" in "covariant" meant with - something that varies with, not against, what you're doing. But later someone started using "co-" to mean, umm, "contra-".

John Baez (Dec 01 2020 at 20:26):

Who first started saying stuff like "coalgebra"?

Reid Barton (Dec 01 2020 at 20:26):

Just think that we could have had the infinitely more steampunk "contralimits", "contramonads" etc.

John Baez (Dec 01 2020 at 20:27):

That would be great. But we're in so much of a mess now there's no way to get out, short of a civilizational reboot.

Jules Hedges (Dec 01 2020 at 20:30):

John Baez said:

Who first started saying stuff like "coalgebra"?

My guess would be Eilenberg and Mac Lane first named colimits, coproducts etc

Christoph Thies (Dec 01 2020 at 20:30):

John Baez said:

Who first started saying stuff like "coalgebra"?

Doesn't it make sense as the notion comes with that of an algebra?

James Wood (Dec 01 2020 at 20:31):

Jules Hedges said:

John Baez said:

Who first started saying stuff like "coalgebra"?

My guess would be Eilenberg and Mac Lane first named colimits, coproducts etc

What about “cosine”?

Dan Doel (Dec 01 2020 at 20:34):

According to an overflow answer, it means something like 'sine of the complementary angle'.

Jules Hedges (Dec 01 2020 at 20:35):

Interesting, I wonder if "co" in category theory is short for complementary, that would actually make sense

John Baez (Dec 01 2020 at 20:35):

Re "coalgebra":

Doesn't it make sense as the notion comes with that of an algebra?

But everyone reads "co-" in "colgebra" to mean the "like an algebra but backwards, or upside-down". In other words, "co-" indicates a reversal.

I think "cosine" is a good clue.

John Baez (Dec 01 2020 at 20:37):

Which came first, "colimit" or "coalgebra"?

Jules Hedges (Dec 01 2020 at 20:38):

My guess would be colimit, does it appear in the very first Eilenberg-Mac Lane paper?

Oscar Cunningham (Dec 01 2020 at 20:38):

It would work if we just switched to saying that functors were covariant or 'variant'.

John Baez (Dec 01 2020 at 20:38):

The physicists and geometers were doing pretty well using "co-" and "contra-" as antonyms.

John Baez (Dec 01 2020 at 20:38):

But now things are hopelessly fuddled.

Christoph Thies (Dec 01 2020 at 20:45):

John Baez said:

But everyone reads "co-" in "colgebra" to mean the "like an algebra but backwards, or upside-down". In other words, "co-" indicates a reversal.

Isn't this what it is? A co-coalgebra is an algebra, isn't it?

Nathanael Arkor (Dec 01 2020 at 20:54):

Jules Hedges said:

My guess would be colimit, does it appear in the very first Eilenberg-Mac Lane paper?

Limits and colimits used to be called inverse limits and direct limits, so "co-" in "colimit" appeared relatively late.

Jules Hedges (Dec 01 2020 at 20:55):

Christoph Thies said:

Isn't this what it is? A co-coalgebra is an algebra, isn't it?

No. An algebra is a map $F(X) \to X$ for some functor, a coalgebra is a map $X \to F(X)$

John Baez (Dec 01 2020 at 20:56):

Christoph Thies said:

John Baez said:

But everyone reads "co-" in "colgebra" to mean the "like an algebra but backwards, or upside-down". In other words, "co-" indicates a reversal.

Isn't this what it is?

Yes, that's what it is. I'm saying that in the term "coalgebra", people don't read "co-" as meaning "with", contrary to what you seemed to be suggesting. They read it as meaning "opposite to".

John Baez (Dec 01 2020 at 20:57):

A modern mathematician would read "cooperation" and think "operating in reverse".

John Baez (Dec 01 2020 at 20:58):

Which is why if you ask a mathematician to cooperate they'll do the exact opposite of what you want.

Eric Forgy (Dec 01 2020 at 21:15):

To get anything done, you need copsychology.

Amar Hadzihasanovic (Dec 01 2020 at 21:31):

My guess is that the original source of the “co-” prefix is the pair domain/codomain, since the most obvious effect of passing to the opposite category is that these two are reversed.
According to this thread, “domain/codomain” seems to appear in 1922 for the first time, and be based on Russell&Whitehead's terminology of “domain/converse domain” in the Principia Mathematica.

Amar Hadzihasanovic (Dec 01 2020 at 21:34):

So our categorical “co-” would originate from converse.
That's a different “co-” from the one in cosine, which seems to be an abbreviation of sine of the complementary angle (sinus complementi -> cosinus).

Jules Hedges (Dec 01 2020 at 21:42):

There's also the converse of a relation

Amar Hadzihasanovic (Dec 01 2020 at 21:43):

Interestingly, in “converse” the “con-” is not really doing much semantic work, it just strengthens the word -- “versus” means “turned”, “conversus” means “turned around completely”

Jules Hedges (Dec 01 2020 at 21:43):

So the modern terminology exactly matches Russell & Whitehead: the codomain of a function is its domain in the opposite category

Amar Hadzihasanovic (Dec 01 2020 at 21:45):

So maybe, really, we should have used “versi-” as a prefix... :D

Jules Hedges (Dec 01 2020 at 21:45):

versilimit, versialgebra

Amar Hadzihasanovic (Dec 01 2020 at 21:49):

Tbh, a proper Greek/Latin prefix which means 'opposite' does exist: it's just “anti-”.

Nathanael Arkor (Dec 01 2020 at 21:50):

There are trigonometric functions called "versine" (1 - cosine) and "coversine" (1 - sine): the geometers are really making the most of the different prefixes.

Jules Hedges (Dec 01 2020 at 21:51):

Instinctively I think "antilimit" sounds kinda reasonable, but "antialgebra" sounds very strange

Amar Hadzihasanovic (Dec 01 2020 at 21:51):

I think we really missed an opportunity with “antialgebra”, it sounds like something which is there to wage a war on algebra.

Amar Hadzihasanovic (Dec 01 2020 at 21:58):

The next step is to abolish compounds that mix words of Greek origin with Latin prefixes and viceversa (polycategory: yay, multicategory: nay)...

Jules Hedges (Dec 01 2020 at 22:05):

What about mixing with Arabic words like "algebra"?

Ian Coley (Dec 01 2020 at 22:11):

Or Germanic/old English words like "ring"?

Fabrizio Genovese (Dec 01 2020 at 22:16):

Amar Hadzihasanovic said:

The next step is to abolish compounds that mix words of Greek origin with Latin prefixes and viceversa (polycategory: yay, multicategory: nay)...

This. Mixing Latin and Greek in naming things really is an unholy practice.

Ian Coley (Dec 01 2020 at 22:45):

Fabrizio Genovese said:

Amar Hadzihasanovic said:

The next step is to abolish compounds that mix words of Greek origin with Latin prefixes and viceversa (polycategory: yay, multicategory: nay)...

This. Mixing Latin and Greek in naming things really is an unholy practice.

I guess this makes you against both homosexuality and heterosexuality (but not bisexuality!). But this is a silly hill to die on. So many of the tiny prefixes and suffixes (re-, a-, -er, -ist) are Latin or Greek affiliated and these are never seen as an issue.

Fabrizio Genovese (Dec 01 2020 at 23:07):

Ian Coley said:

Fabrizio Genovese said:

Amar Hadzihasanovic said:

The next step is to abolish compounds that mix words of Greek origin with Latin prefixes and viceversa (polycategory: yay, multicategory: nay)...

This. Mixing Latin and Greek in naming things really is an unholy practice.

I guess this makes you against both homosexuality and heterosexuality (but not bisexuality!). But this is a silly hill to die on. So many of the tiny prefixes and suffixes (re-, a-, -er, -ist) are Latin or Greek affiliated and these are never seen as an issue.

It's actually more complicated than this. The rule applies when you are taking two words that do not exist in your language and mix them up. "Homosexuality" is fine because "sexuality" is already a well-established English word.

Fabrizio Genovese (Dec 01 2020 at 23:09):

What I'm talking about applies mainly in fields like biology, where you take words directly from Latin/Greek, such as "Aedes albopictus" or "Bubo bubo".

Fabrizio Genovese (Dec 01 2020 at 23:09):

So, for instance, "pericardiac" is perfectly acceptable. "Circumcardiac" is just next-level horrible.

John Baez (Dec 02 2020 at 00:06):

Amar Hadzihasanovic said:

My guess is that the original source of the “co-” prefix is the pair domain/codomain, since the most obvious effect of passing to the opposite category is that these two are reversed.
According to this thread, “domain/codomain” seems to appear in 1922 for the first time, and be based on Russell&Whitehead's terminology of “domain/converse domain” in the Principia Mathematica.

Nice!!!

Ian Coley (Dec 02 2020 at 00:54):

Well 'category' is already a well-established English word, so I don't think you can reject multicategory. Or you could call them 'coloured operads' and avoid the trouble. (Although looking at nLab they might not be exactly the same ... so you'd have to call them non-symmetric coloured operads)

Reid Barton (Dec 02 2020 at 00:57):

shouldn't they be asymmetric? :upside_down:

Fabrizio Genovese (Dec 02 2020 at 02:12):

Ian Coley said:

Well 'category' is already a well-established English word, so I don't think you can reject multicategory. Or you could call them 'coloured operads' and avoid the trouble. (Although looking at nLab they might not be exactly the same ... so you'd have to call them non-symmetric coloured operads)

Indeed I wasn't referring to the "polycategory" issue specifically, but to the general practice :smile:

Ian Coley (Dec 02 2020 at 02:31):

Reid Barton said:

shouldn't they be asymmetric? :upside_down:

:silence:

Jason Erbele (Dec 02 2020 at 19:01):

Fabrizio Genovese said:

Ian Coley said:

Fabrizio Genovese said:

Amar Hadzihasanovic said:

The next step is to abolish compounds that mix words of Greek origin with Latin prefixes and viceversa (polycategory: yay, multicategory: nay)...

This. Mixing Latin and Greek in naming things really is an unholy practice.

I guess this makes you against both homosexuality and heterosexuality (but not bisexuality!). But this is a silly hill to die on. So many of the tiny prefixes and suffixes (re-, a-, -er, -ist) are Latin or Greek affiliated and these are never seen as an issue.

It's actually more complicated than this. The rule applies when you are taking two words that do not exist in your language and mix them up. "Homosexuality" is fine because "sexuality" is already a well-established English word.

By that standard, "television" might get a pass because "vision" was already well established in the English language when the boob tube was introduced, but "Wikipedia" should be abolished.

Fabrizio Genovese (Dec 02 2020 at 19:45):

Indeed. :smile:

Jules Hedges (Dec 02 2020 at 19:46):

I think encyclopaedia has been around for a while

Jules Hedges (Dec 02 2020 at 19:49):

(Long enough for the American spelling to have time to diverge from the English spelling at least)

Fabrizio Genovese (Dec 02 2020 at 19:50):

Yes, but "pedia" is not a word per-se. Indeed it comes from παῖς, -ός which means "kid"

Fabrizio Genovese (Dec 02 2020 at 19:52):

To be absolutely precise, "pedia" comes from παιδεία, "education of the kids", which itself comes from παῖς. :smile:

Morgan Rogers (he/him) (Dec 02 2020 at 20:53):

Or one can just enjoy the flexibility of language in assembling pieces from different roots :shrug: There's clearly no basis for a 'consistent roots' rule of word formation in English anyhow.

Fabrizio Genovese (Dec 02 2020 at 21:09):

In taxonomy it's a rule, actually. :smile:

Fabrizio Genovese (Dec 02 2020 at 21:11):

In any case, everyone is entitled to their personal preference. There are paths also in cross-language contamination, and having personally always seen the world from a linguistic lens I find these kind of neologism quite unpleasant.

Morgan Rogers (he/him) (Dec 02 2020 at 21:34):

What's a computer called in Italian, again? :stuck_out_tongue_wink:

Morgan Rogers (he/him) (Dec 02 2020 at 21:35):

It's not like you were born before they proliferated; where does that feeling of unpleasantness come from, do you think?

Fabrizio Genovese (Dec 02 2020 at 21:43):

[Mod] Morgan Rogers said:

What's a computer called in Italian, again? :stuck_out_tongue_wink:

That is a completely different issue, you are mixing things up. My personal prefernce comes from the fact that this has been seen as an issue in intellectual circles since roughly the XVII century, it's not something I'm making up out of nowhere. You can use this as reference. https://en.wikipedia.org/wiki/Barbarism_(linguistics)

Fabrizio Genovese (Dec 02 2020 at 21:45):

(I've been trained in Classics and I'm fluent in Latin, not fluent but I can translate from Ancient Greek. This makes me probably more sensitive to this issue.)

Morgan Rogers (he/him) (Dec 02 2020 at 22:09):

Fabrizio Genovese said:

this has been seen as an issue in intellectual circles since roughly the XVII century

I could surely read that as "there has been language snobbery for centuries". What I want to know is why, personally to you, does the fact that you can recognise when a word is constructed from different roots lead to a feeling of unpleasantness? It's not like these words are being used in their original languages. And why are modern borrowings, or derivatives from them, not comparable?

Matteo Capucci (he/him) (Dec 02 2020 at 22:12):

To use a Latin expression, de gustibus :stuck_out_tongue_wink:

Chad Nester (Dec 02 2020 at 22:13):

To get us back on (off?) track, what about “abacklimits”.

Chad Nester (Dec 02 2020 at 22:14):

“A sail is aback when the wind fills it from the opposite side to the one normally used to move the vessel forward.”

Chad Nester (Dec 02 2020 at 22:15):

... “foreboundary”, “aftboundary”

Chad Nester (Dec 02 2020 at 22:15):

Port and starboard adjoints.

Reid Barton (Dec 02 2020 at 22:16):

leeward kan extension

Fabrizio Genovese (Dec 02 2020 at 22:35):

Chad Nester said:

To get us back on (off?) track, what about “abacklimits”.

Yes please, I've had more than enough of this.

Chad Nester (Dec 02 2020 at 22:48):

$\mathbb{C}^{op} \to \mathbb{D}$ is “fore-aback”, $\mathbb{C} \to \mathbb{D}^{op}$ is “aft-aback”.

Chad Nester (Dec 02 2020 at 22:51):

In higher categories the n+2-structure is clearly “the rigging”.

Chad Nester (Dec 02 2020 at 22:55):

... the obvious adjective seems to be “tangled”, so I guess for 2-categories $\mathbb{C}^{co} \to \mathbb{D}$ has “tangled fore-rigging”, and so on.

Chad Nester (Dec 02 2020 at 23:25):

I guess it makes sense to shorten this to “fore-tangled” and “aft-tangled”... so to answer the original question a 2-functor $\mathbb{C}^{co} \to \mathbb{D}$ should be called a “fore-tangled 2-functor”.

Chad Nester (Dec 02 2020 at 23:26):

A major advantage of this taxonomy is that you get to draw a sailboat whenever you explain it in your papers.

Chad Nester (Dec 03 2020 at 11:55):

Okay so above "fore" and "aft" are mixed up. Oops!

One more nautical terminology suggestion... (after Reid Barton's above)

"Windward is the direction upwind from the point of reference, alternatively the direction from which the wind is coming. Leeward is the direction downwind from the point of reference."

We might call both limits and colimits as just "limits". Then what we usually call limits are "windward limits", and colimits are "leeward limits". This is intuitive if you think about (co)cones: In a windward cone for a diagram the arrows are all going towards the diagram in question, while for a leeward cone they are going away from it (like wind direction). Then a limit of a diagram is a "windmost cone", and a colimit of a diagram is a "leemost cone" (or something).