I’ve finished Radishchev’s Journey (see this post), and the final chapter, on Lomonosov, was actually reasonably interesting. In the course of discussing Lomonosov’s thirst for learning, acquired in part through mastery of foreign languages, Radishchev writes (I quote the Leo Weiner translation, published by Harvard University Press, 1958):
Thus the student, upon approaching an unknown language, is confused by strange sounds. His throat is exhausted by the unfamiliar rustling of air escaping from it, and his tongue, compelled to wag in a new way, grows lame. The mind grows stiff, reason is weakened by inactivity, imagination loses its wings; memory alone is wide awake and ever keener, filling all its convolutions and openings with hitherto unknown sounds. In learning languages, at first everything is repulsive and burdensome. If one were not encouraged by the hope that, after having accustomed his ear to the unusual sounds and having mastered the strange pronunciation, most delightful ways would be opened up to him, it is doubtful that one would want to enter upon so arduous a path. But when these obstacles are surmounted, how generous is the reward for perseverance in overcoming hardships! New aspects of nature, and a new chain of ideas then present themselves. By acquiring a foreign language we become citizens of the region where it is spoken, we converse with those who lived many thousand years ago, we adopt their ideas; and we unite and co-ordinate the inventions and the thought of all peoples and all times.
And, approaching the end of Slezkine’s wonderful Arctic Mirrors: Russia and the Small Peoples of the North
There is no point in writing in my native language because out of eight thousand Nanai living in this world, if anybody reads poetry, they read it in Russian. There is no need to translate Pushkin into Nanai. I love Pushkin in the element of Russian speech and I cannot reject it. In any case, writing poetry in any other language strikes me as strange. And who knows to what extent Russian has become my native language?
This is from Идти вечно [To always go/walk] (1972), which Slezkine calls Kile’s “remarkably fresh memoir.”
I assume “Kile” has the stress on the second syllable—anybody know?):
«ударение […] всегда падает на последний слог слова, независимо от его морфологической структуры.» (Аврорин, 1959)
Спасибо!
I particularly identify with the first quote, I believe that the most effective single means by which you can understand a people, a culture, a country, is to learn their language (and then, of course, use it to actually communicate with those people). Never forget that the true purpose of language is communication, it’s not really there to serve as an interesting little mental puzzle for you to solve, though some people may treat it as such and that’s fine.
Cheers,
Andrew
I disagree. I think that the true purpose of language is to serve as an interesting little mental puzzle for me to solve. Of course some people may treat it a means of communication, and that’s fine.
Cheers,
Empty
I agree. I think the true purpose of communication is to pose mental puzzles in the form of language. As Talleyrand remarked: “Language was given to man to disguise his thoughts”.
Blasphemy! Languages were created to confuse and scatter humankind so that they couldn’t build a tower to heaven, and that’s that. Moses also spoke all the Hebraic dialects in the Pentateuch, BTW.
Random etymological fact: Spanish lástimar ‘injure, damage, offend, pity’ is a semicultismo, or learned word with popular modifications < blasphemare.
I believe there shouldn’t be a stress mark in lastimar. This is the infinitive, and has the default stress on the final syllable. The noun lástima, though, needs a stress mark over the first “a”, since writing it without any stress mark would indicate that it is to be pronounced *lasTIma. In fact it is pronounced LAStima, as I happen to know from the everyday exclamation Qué lástima ! = “what a pity”.
Jim: Blasphemy! Languages were created to confuse and scatter humankind so that they couldn’t build a tower to heaven, and that’s that.
At last: somebody here who can provide discussion-free guidance about languages ! What is the correct Christian position regarding skyscrapers ? My inclination is to avoid them in future – look what happened to the Twin Towers.
I hear they’re building one in Dubai so tall that those in the penthouse will be able to reach up and strike the firmament with hatchets until its waters flow forth. That’s no good. There may soon be several different languages in that region, and I wouldn’t be a bit surprised if Google Translate suffered mysterious technical problems.
Yes, thanks, Grumbly. I accidentally copied the stress from lástima.
My Spanish consultant says that the verb is rare in any sense except ‘injure’, and even that is found mostly in Galicia and America; the rest of Spain says hacerse daño ’cause damage’ or a more general verb like doler. There are a few fixed phrases like su honor lastimado ‘your offended honor’. Such are the perils of relying on dictionaries: they don’t sort out the ordinary from the unusual senses.
James: I thought it was going to be in Iraq, at Tell el-Muqayyar.
There is only one true language. Are there any very tall buildings in the Low Countries?
James: I thought it was going to be in Iraq, at Tell el-Muqayyar.
As you know, John, history doesn’t repeat itself — it rhymes. Speaking of which, they’ll probably go at it with chainsaws, not hatchets. That would explain rising sea levels better than the sparse evidence for global warming!
There is only one true language.
One upped again, you bastard!
JC: hacerse daño ’cause damage’
I think this means ’cause damage to oneself, hurt oneself’, as opposed to hacer daño (a …) ’cause damage, hurt (someone)’.
Similarly in French: se faire mal ‘to hurt oneself’ (by falling, cutting oneself, etc by accident or deliberately) vs faire mal (à quelqu’un) ‘to cause damage to, hurt (someone)’. If for instance you bump your elbow against a hard object, you can say Ça fait mal ‘It hurts!’
Apparently hacerse daño can mean either ‘hurt oneself’ (reflexive), ‘be hurt’ (pseudo-passive), or ‘hurt’ (pseudo-passive with active sense), at least in Spain.
Such active semantics are not uncommon in Spanish: reír and reírse both mean ‘laugh’ (though *reír de whereas reírse de ‘laugh at’); piénsalo ‘think about it’, piénsatelo ‘think about it hard/thoroughly/for a while’. Some semantic shifts are even more drastic: combinar ‘combine’ vs. combinarse (in the plural) ‘take turns’; acusar ‘accuse’ but acusarse not ‘be accused’ but ‘confess’ (shades of the Inquisition)!
JC, there are similar things in French: rire de ‘to laugh about’ (as a spontaneous reaction) and se rire de ‘to laugh mockingly about, not to take s. seriously’ (as an attitude)(the latter is an older, more literary phrase). Quite often the plain verb designates a plain action (or reaction) while the pronominal verb adds a connotation of duration or habit.
Why should Spanish acusarse mean ‘to be accused’ rather than ‘to accuse oneself’? In French too, during confession (to a Catholic priest) you are supposed to say Je m’accuse (de …) lit. ‘I accuse myself’, meaning ‘I confess, I take the blame (for … [having sinned])’.
Grumbly: Perhaps confess …self in the quotations are calques (literal translations) of French se (etc) confesser ‘to confess one’s sins’ (during confession to a priest).
Why should Spanish acusarse mean ‘to be accused’ rather than ‘to accuse oneself’?
Yes, I saw that as soon as I hit the Submit button — which indeed does inspire submission to the inexorable workings of the Machine. I once saw a cartoon of an IRS (U.S. tax collector) office with the auditing of a fearful taxpayer well under way and the famous quatrain from Fitzgerald’s Rubaiyat framed on the wall:
The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.
Singularly apropos.
@John Cowan: in my experience (native Rioplatense speaker, 10+ years in Spain), acusarse ‘confess’ is obsolete, and combinarse ‘take turns’ is strictly a Caribbean/Mexican usage; neither would be acceptable in any of the dialects I speak.
It’s long been my opinion that speaking of ‘passive voice’ in a Spanish context is a misnomer. The be+VBN syntax usually designated by this term is just one of several semanticall-equivalent alternatives, and by no means the most frequent; in spoken discourse, the pronominal construction is much more common (e.g., se construyó instead of fue construido). In the present tense, in fact, be+VBN is usually unavailable.
Alon: Thanks. How much have you consciously adjusted your dialect since moving to Spain? I’m curious.
@John: I codeswitch. I speak (a passable imitation of) the Castilian standard to Spaniards, and my native Rioplatense when back at home.
A few months in Salamanca soon after I’d moved to Spain convinced me that using any dialect but the local standard was about as communicatively effective as speaking English or French— perhaps useful for short-term transactional purposes but a disabling barrier to any kind of long-term relationship.
Fascinating! I’d never have guessed it would be that much of a problem.
In learning languages, at first everything is repulsive and burdensome. If one were not encouraged by the hope that, after having accustomed his ear to the unusual sounds and having mastered the strange pronunciation, most delightful ways would be opened up to him, it is doubtful that one would want to enter upon so arduous a path.
This doesn’t really apply to people who enjoy languages for their own sake (i.e. me and probably most other readers of this blog). In Lavengro, George Borrow took a dim view of this focus on linguistic form rather than meaning:
Borrow was full of it. Exhibit A: Wild Wales.
I don’t disagree. The last paragraph of Wild Wales:
Good lord, shades of Brichot:
(A small sample; there’s pages and pages of this stuff.)
and why has the Welsh only four words for a hill, and its sister language the Irish fifty-five?
Irish words for a hill, Eskimo words for snow…
It’s probably because Wales is notoriously flat, whereas Ireland is famous for its frightful precipices and Horrid Gulphs.
Irish words for a hill, Eskimo words for snow…
Japanese words for cardinal numerals…
No wonder Japanese technology is so advanced!
@de
Wales:
Minimum elevation: 0 ft
Maximum elevation: 3,376 ft
Average elevation: 266 ft
Ireland:
Minimum elevation: 0 ft
Maximum elevation: 3,898 ft
Average elevation: 151 ft
That’s the joke.
As a native speaker of Danish, the claim that Spanish doesn’t use its ‘passive’ very much seems a bit off — we only have the mediopassive in -s so allí se habló danés and danés fue hablado allí are unproblematically equivalent (and se encuentran = de mødes is middle, ‘they meet each other’, but still the same thing). Maybe fixating on the difference(s) is an Anglophone bias?
Maybe fixating on the difference(s) is an Anglophone bias?
What else ? Once you have heard enough Spanish, there’s nothing left to fixate on. It’s all a blur of intelligibility. True of every language I’ve learned, of course, including English.
It appears to be mostly amateur linguists who get hypnotized, say on “participles come at the end of sentences !!”.
Al final de las oraciones aparecen … los participios !!
¡Olé!
I thought it was Italy that was famous for its Horrid Gulphs and Ghastly Gibbelins (who eat, as is well known, nothing less good than man).
“From all the hiding places come the Indians”, a sentence used to illustrate the use of CH₂O.
Wow. I can answer like half of them off the top of my head.
The philosopher that comes to mind is Otto. Don’t call him stupid.
To be strictly fair, Borrow did say human reason.
Ape!
Absolutely. Both DM and I are tellurian primates!
I just looked up tellurian in the OED, since its appearance here reminded me that I had been curious about its frequency in relation to the seemingly much more common Terran in science fiction. Besides the oddity that one is (usually) considered a proper adjective and the other not, I discovered a few things. I knew that the Latin etymons terra and Tellus* were both words for the Earth, although they are unrelated, and Tellus was in classical times pretty much restricted to the personified Earth divinity (who could be female or male, depending on circumstances). However, I also learned that Tellus was derived from a root meaning “floor” and so it too must presumably have been a common noun originally—just conceiving of Earth in a different way from terra, which is derived from meaning of “dirt” (and before that “dry stuff,” presumably as opposed to the hydrosphere).
As to the occurrence of tellurian in science fiction, the OED has only one post-1954 citation for the adjectival form (although more for the noun), which caught my attention because of the ongoing discussion of small insects:
1997 I. S. Behr & R. H. Wolfe Legends of Ferengi 148 The workers have a life expectancy only slightly longer than a Tellurian gnat.
However, the ultimate reason that I am posting anything here at all is that I discovered another definition of tellurian:
†2. Applied by the political writer John Reeves to anti-monarchists. Obsolete. rare–1.
As part of an extended figure of speech likening anti-monarchist or republican views to the geocentric view of the universe.
The OED’s only cites for this are Reeves himself, once for the word as an adjective and once as a noun.
1799 J. Reeves Thoughts Eng. Govt. III. 65 The Tellurian Politicians..would take the Sovereign from his Sphere, and make him something like an attendant upon his subjects in parliament.
1799 J. Reeves Thoughts Eng. Govt. III. 66 The unscientific Sectarists, your friends the Tellurians, were presently up in arms.
* Amusingly, which one is a usually a proper noun here is inverted from the case of the modern English adjectives.
As it happens, the word itself is neuter.
Feminine.
Nunc est bibendum
Nunc pede libero
Pulsanda tellus.
Brett: Tellus was derived from a root meaning “floor”
Old Norse:
þili n. “wooden cladding, wooden wall, board floor”
þilja f. “deck plank, floorboard”
ON þilir m.pl. “men from Telemark”, obviously “Tellurians”
“men from Telemark”, obviously “Tellurians”
For the most part, yes.
What all that heavy water was really for …
@trond
Given the age of the word, I wonder if the original sense was not “flat earthen floor for threshing grain”. That is how I first encountered Diele in German.
The real naming problem is with people (not necessarily human) from Venus. Venusian is a barbarism; Venerean is analogous to Martian, Jovian, etc., but too close to venereal for comfort; Cytherean is ideal except for being hard to recognize.
In my revision of “Omnilingual”, the planet isn’t Mars-as-we-know-it, so I chose to use Ares, Arean; Harry Turtledove called his only somewhat similar planet Minerva, giving Minervan. Ares has an atmosphere similar in the lowlands to Earth’s highlands, so supplementary oxygen is all you need; Minerva is Earth-sized with a full oxygen-nitrogen atmosphere, so the greenhouse effect keeps it warm, if chilly by Earth standards.
Aphroditic?
OK, perhaps not.
Why not pulsandam, then? Or is it “now I, the earth that is to be kicked, am liberating someone or other”?
No: it represents Pulsanda [est] tellus “the earth is to be kicked”, carrying over the est from Nunc est bibendum. It’s a gerundive. “Now it’s to-be-drunk; now, with a free foot, the earth is to-be-struck.” The sense can be summed up as “Toga! Toga! Toga!”
The poem (as it carries on) is both misogynistic and xenophobic, and moreover a toadying piece of naked political propaganda. It’s also a terrific poem. I don’t like Horace, but mehercle, he was some poet.
Cf. Carthago delenda est.
Oh, I didn’t consider the possibility of a free foot.
This provides me with a flimsy yet exploitable pretext for citing the end of my favouritest Horatian ode EVAH (“Eheu fugaces”), the first line of which is confronting me just now, inscribed on an eighteenth-century clock repaired by my father and given to us as a wedding present (presumably in a memento mori sort of spirit.) The poet enjoins what one must say is a somewhat Republican (if not outright Bolsonarian) approach to the coronavirus situation, concluding
Linquenda tellus et domus et placens
uxor, neque harum quas colis arborum
te praeter invisas cupressos
ulla brevem dominum sequetur;
absumet heres Caecuba dignior
servata centum clavibus et mero
tinguet pavimentum superbo,
pontificum potiore cenis.
What is it with that cypress? I for one never trusted the buggers, hanging around in graveyards and all, but I didn’t know it went back that far…
And better wine than the Pontiff, not bad for a poet’s cellar.
(I’m torn, do I try to understand Kan extensions or learn to construe Latin verse — I had to cheat).
I would have thought that anybody who can understand Kan extensions would find construing Latin verse a bit of a doddle. Good for unwinding at the end of a hard day extending Kans.
not bad for a poet’s cellar
Horace was one of those successful poets. We all know the type …
There’s a useful analysis (and Latin prose version, helpful to us amateurs) here, as well as a lively translation and this meditation on the gerundive:
Kan extensions can’t be as bad as Kant extensions.
@Lars: Kan extensions
Even the WiPe has wise words on them’uns:
# Kan extensions generalize the notion of extending a function defined on a subset to a function defined on the whole set. The definition, not surprisingly, is at a high level of abstraction. When specialised to posets, it becomes a relatively familiar type of question on constrained optimization. #
If you want to talk at that “high level”, prepare for life as a hermit. Merely thinking at that level is fine, just don’t be surprised that you must keep it for yourself and a few other select hermits.
This message is brought to you by an enthusiast of the Luhmann brand of Advanced Thinking.
It seems to me that the force of the gerundive can be approximated very well by a modifying infinitive. A wave to sail, the Cocytys to gaze upon, the earth to leave behind. English also has a sort of perfect anti-gerundive, “to be done but wasn’t: a/the road to take -> the road not taken.
I should dig out Saunders and MacLane from the garage, I’m pretty sure I have it. They have a whole chapter on the buggers, it seems. This time I started with the closure of a poset and went up in abstraction instead of drawing a sketch, I have only myself to blame.
And I promise I shan’t attempt to explain them here.
We don’t have any construction in English that carries the full force of the gerundive
This sort of thing annoys me. We don’t have anything that carries the full force of the ablative absolute, either. So what? Latin lacks the near-supernatural linguistic power of our “definite article.”
To attribute the power of the ode to the fact that Latin has a gerundive is to miss the point pretty spectacularly. The effect is due to Horace’s masterly repetition and placement of those gerundives, i.e. to the fact that he’s a great poet. When it comes to language and poetry, it ain’t what you got, it’s what you do with it that counts.
The gerundive can just as well be tremendously leaden (as Cato was doing on purpose with his Carthago delenda est every damn time he spoke in the Senate.)
I promise I shan’t attempt to explain them here.
Damn. There was I hoping that you would.
This sort of thing annoys me. We don’t have anything that carries the full force of the ablative absolute, either. So what? Latin lacks the near-supernatural linguistic power of our “definite article.”
And it would be perfectly reasonable to point that out in dealing with an English poem that made powerful use of the article (if one were thinking of translating it into Latin, of course). I think you may be overreacting; I didn’t at all get a sense of “See how supernaturally subtle Latin is, you English-speaking peons?” but rather “Latin has this feature that English doesn’t, which can make things harder for English-speakers.”
The gerundive can just as well be tremendously leaden (as Cato was doing on purpose with his Carthago delenda est every damn time he spoke in the Senate.)
I’m not sure what you mean by “leaden,” but the fact that it was a tremendously effective slogan that is remembered to the present day suggests it worked pretty damn well. (And I am no fan of Cato’s.)
What is the correct Christian position regarding skyscrapers ?
Luke 14:28-30, probably.
“For which of you, intending to build a tower, sitteth not down first, and counteth the cost, whether he have sufficient to finish it?
Lest haply, after he hath laid the foundation, and is not able to finish it, all that behold it begin to mock him,
Saying, This man began to build, and was not able to finish.”
(This section is one of my favorite parts of the entire Bible, for its sheer timeless practicality. Alas, far too many would-be tower-builders of the modern day have seemingly forgotten it.)
January’s theological statement strikes me as definitive.
Yes, that is now officially one of my favorite parts of the Bible as well. Thanks for giving it the prominence it deserves.
The man who counteth the costs and decides to cut them will finish many shoddy buildings. He will also buy many properties and let them deteriorate while raising the rents. He will be mocked, then elected president. [Theophrastus Bombast von Hohenheim, passim]
I was idly considering, for my next blockbuster, a work demonstrating that Donald Trump is in fact the Antichrist (it all fits, I tell you!)
Unfortunately the main market for books about the imminent fulfilment of the prophecies in the Book of Revelation seems to consist largely of people who seem unaccountably unsympathetic to my conclusions. Truly, Biblical scholarship is at a low ebb in these end times.
So effective that Cato didn’t have to say it for it to be remembered.
Now David (not M) has me worried, maybe Kan extensions are so abstract because every time somebody is able to explain them some giggling guy from Alpha Centauri replaces them with something even more abstruse.
No, not at all. You really mustn’t worry about that.
That’s all I needed to hear, coming from you.
From Wikipedia: “The notion of Kan extensions subsumes all the other fundamental concepts of category theory.”
The good news: if you understand Kan extensions you understand all fundamental concepts of category theory.
The bad news: In order to understand Kan extensions, you have to understand all fundamental concepts of category theory.
More like: for any given concept of category theory, if you don’t need to understand it to understand Kan extensions, it is not fundamental.
Category theorists are actually very explicit about avoiding circularity of concepts — you will find remarks about how ‘basic’ concepts and constructions can be seen as special cases of more abstract ones but it’s unclear how to get to the latter without using the former, so you have to have both.
Kan extensions seem to be one of those cases — they may ‘subsume’ a lot of other constructions, but it’s not much use unless you take them on faith. And you can’t make them into axioms because they only exist if they do.
Originally, the amount of category theory you could understand, and profit from understanding, was directly proportional to the number of different fields of mathematics in which you had scrumped apples. Its purpose was that of striptease – to peel away the veils of karma, revealing the naked essentials.
Now it’s a show in its own right, with smoke and mirrors instead of moonlighting housewives..
Learn not, want not.
Moonlighting housewives? Why was I never told?
Housewives to moonlight. If it can’t be done gerundively, don’t do it.
And I think the term ‘moonshine’ is common with Kan extensions.
I did not expect to find King Ghidorah in there. What next, the Spanish Inquisition?
This kind of thing is why pure math always defeated me. You can’t understand it properly bottom-up, and you can’t understand it properly top-down either. You just have to grok the whole thing at once.
Not all at once. Run bottom up for a while, then run top down for a while, repeat. Back and forth between “practice” and “theory”. Like learning a language.
In most cases you have to be more than merely “interested in learning”. You must have a dog in the fight, even if it’s only a stuffed one.
When I was in college my dog lost its stuffing and I had to leave the math department.
I read Hat’s comment before Stu’s. My initial reaction was simply one of great sadness.
You must have a dog in the fight
True. If I had to learn some piece of math in order to fully grasp how to make use of some fancy piece of theoretical physics, I could usually knuckle down and do it. Understanding math for its own sake was never sufficient motivation.
It was said of the Report on the Algorithmic Language Algol 68 that you could not understand any part of it without already understanding all of it. The Revised Report improved the situation: you could understand each part deeply if you understood all other parts to at least some degree. The RR’s grammar is unique: it specifies not only the syntax of the language and its static semantics (semantics that can be checked before the program starts), but also its dynamic semantics.’
The stuffed dog reminds me of the teddy bear that student programmers had to explain their difficulties to before they could get human help. It was said to be very effective, and I believe it. I once wrote a simulator for it that would mostly say nothing, but occasionally when spoken to it would emit a random strategy from Brian Eno’s “Oblique Strategies” cards.
I still have my stuffed dog. It can’t do much more nowadays than growl at monads.
The dog growls, the monads move on.
Lars Mathiesen says: As a native speaker of Danish, the claim that Spanish doesn’t use its ‘passive’ very much seems a bit off — we only have the mediopassive in -s
Not that I know anything about Danish, but doesn’t it also have a periphrastic passive with blive?
English also used to have blive or belive, but it’s obsolete.
Even though the original sense is something like ‘remain,’ the periphrasis with blive has a strong aspect of becoming — it can be bleached almost to a pure passive but a change of state is implied, unlike G werden in the same function, though it feels like it is changing in that direction. I wouldn’t call it just ‘passive’ just yet.
If the -s mediopassive is not wanted, you can also rephrase as active with man as the subject.
But in any case, Danish doesn’t select between these constructions in a way that corresponds to a difference (if any) between hablarse and ser hablado.
To make a really old and sickening type of prescriptivist joke: is the claim that Spanish doesn’t use its “passive” very much really a native speaker of Danish?
I don’t get it.
I had As where some might insist on To. (Or I find to be instead of it seems).
Prescriptively, the rule is that a fronted modifier like that should refer to the subject and not to some other NP (“When walking down the street, a large number of books is cumbersome to me”) or some other referent that isn’t even mentioned (“If found guilty, the lawsuit could cost billions”). These are participles, hence the term dangling participle, but any modifier will do.
Pullum says the rule is not a grammatical one — it is too easily and frequently violated in both speech and writing — but a rule of politeness, like saying “you and Mark and me” as opposed to “ego et tu et Marcus” (which explains the terms first, second, third person). It may confuse the listener or make them laugh when that was not the speaker’s intent.
Gotcha!
OK, the context for my TED talk about Kan extensions is the way a category theorist can always abstract whatever they are talking about into a category.
Basic example: The product A x B of two sets. You know what it is, I know what it is. One interesting fact about the product is that if you have two functions f and g from a set C to A and B, it is ‘the same’ as having a function (f,g) from C to A x B, because you can then select the two elements of the results and get back your functions (‘compose with projections’). In fact, the product set is the unique set (up to bijections) that allows you to do that.
So that’s how you define a product in a general category — you talk about objects and morphisms (or arrows) instead of sets and functions, but the definition is the same: the object allowing you to ‘pair’ two morphisms and then get the original ones back. It turns out that products of groups, vector spaces, and what have you can be defined exactly that way (in their own categories, where the arrows are usually what were called homomorphisms back home).
When you start finding out how to transfer all sorts of interesting manipulations into category theory, this kind of thing keeps popping up: The ‘least general’ object that behaves a certain way so that you can construct an arrow ‘through it’ from any other object that behaves that way. We must be able to categorify that!
(Yes, least general. Because you can stand on your head and do co-products, which are ‘more general’ than other things).
The trick is to encode the behaviour you want in a diagram which is really a category with ‘abstract’ objects and arrows. For the product, it is just a category with two objects and nothing else, and then picking two sets is a ‘functor’ from that category into Set (the category of sets). And the product itself is encoded as a functor from the one-object category * into Set along with a ‘natural transformation’ to the original functor — which is just another way of saying a set of arrows in Set that commute with the ones in the image of the first functor — while the business about being the ‘least general’ becomes a constraint on the natural transformation.
The second functor along with the natural transformation is then the ‘limit’ of the first functor. One interesting fact is that there is a functor (necessarily unique) from the diagram to *, and that composes with the limit we just defined to give another functor from the diagram category to Set. (In fact that composition is needed to be able to have a natural transformation, I glossed that over).
But hey, where did * and composing with the unique functor just come from? What if we used some arbitrary category and functor in that place, and tried to find a ‘least general’ functor and natural transformation to complete the same picture, just for the hell of it? That’s the Kan extension. (The left one, I think. The right one will make co-products).
And as the man said, there are many other things you do in category theory that can be mapped to the same picture. If you take an identity functor as the basis, you are making left and right adjoint functors instead.
Let me just add a little circularity to this. The functors that are ‘candidates’ to being the Kan extension for a given problem can be organized as a category where the objects are functors each paired with the natural transformation that obeys the constraint (and the arrows are natural transformations as well, with a slightly arcane action on the natural transformation part of the objects). Now of course, the Kan extension we are looking for is an initial object in that category, i.e., a limit over the diagram with no objects. (This may not help you find one, though, it is an extremely abstract fact).
Thanks, Lars. I feel about this just as I do about the long passages of untranslated Russian that Hatters occasionally post: I don’t understand any of it, but I like the flattering implication that I might have done. Or at least, that I hang around with people who do.
You’re welcome.
Category theory is the UG of mathematics, except for the politics.
The horrible thing about it is that even when you think you know in what sense something is a category, you look away for a second and all the arrows turn into objects in their own categories.