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John Baez (Apr 20 2023 at 00:06):I wrote a blog article on Galois connections:
@Jean-Baptiste Vienney might like my treatment of the Nullstellensatz.... even though I don't even state the Nullstellensatz, just an easier theorem!
Jean-Baptiste Vienney (Apr 20 2023 at 00:12):Thank you! This is exactly the stuff I wanted to understand after you told me about the Nullstellensatz as an adjunction.
John Baez (Apr 20 2023 at 00:13):Good! The basic idea is really simple. Then people make it harder.
Jean-Baptiste Vienney (Apr 20 2023 at 00:18):I wanted to understand the list of Galois connections given by @Martin Brandenburg here
Jean-Baptiste Vienney (Apr 20 2023 at 00:19):I will be super happy if I understand all of them, so much space saved in my memory if I understand all of that under a same schema.
John Baez (Apr 20 2023 at 00:19):Nice! I think I know what all those are except "Grothendieck's main theorem of Galois theory". I might know what that is if someone reminds me....
John Baez (Apr 20 2023 at 00:20):A bunch of them are really best thought of as contravariant adjunctions between interesting categories, while some are contravariant adjunctions between mere posets.
John Baez (Apr 20 2023 at 00:24):Furthermore, 3 of them are dualities between commutative rings and spaces, where we get a commutative ring on a space from looking at ring-valued functions on that space. 2 of them are dualities between abelian groups of some kind and abelian groups of the same kind, where we get an abelian group by homming another abelian group into the circle group. And 2 are dualities between 'covering spaces' and their groups of deck transformations.
John Baez (Apr 20 2023 at 00:25):So I'd say there are at most 3 basic ideas kinds of Galois connections listed here... each of which can be worked out in many different ways.
John Baez (Apr 20 2023 at 00:26):If I ever write my book Learning Math With Categories, I should remember this.
David Egolf (Apr 20 2023 at 02:42):Looks very interesting! I'll have to come back to this article on a day when I don't have a headache. Among other things, it looks like this article relates to my ongoing quest to understand adjoint functors...
Well, I had fun understanding a proof of this (from the article):
"On these new smaller versions of the sets X and Y, the maps L and R become inverses! You can prove this simply by fiddling around with the definition of Galois connection. If you’ve never done it before, it takes some thought. "
I tried to prove this on my own, but I eventually had to resort to the nlab. The proof there was a lot simpler than I had expected!
David Michael Roberts (Apr 20 2023 at 04:27):I think what is meant by Grothendieck's Galois theory is the pi_1 version of reconstructing a group from a fibre functor, but here having one valued in sets, not in vector spaces
John Baez (Apr 20 2023 at 05:12):@David Egolf wrote:
I tried to prove this on my own, but I eventually had to resort to the nLab. The proof there was a lot simpler than I had expected!
Yeah, you just have to fiddle with the definitions, but it seems sneaky at first. It helps a lot if you've been told that given order preserving maps
with
we always have
for all and
for all . These foreshadow important facts about adjoint functors, so they're worth remembering! And you can use these to show what we want now, namely
and
dusko (Apr 21 2023 at 01:06):Jean-Baptiste Vienney said:
I will be super happy if I understand all of them, so much space saved in my memory if I understand all of that under a same schema.
it is nice to notice that many mathematical constructions subsume under adjunctions.
but a list of things that are like each other is way less than what lawvere discovered in his 1969 Dialectica paper: that ALL LAWS OF LOGIC ARE ADJUNCTIONS. for the propositional connectives, the observation got expanded in the 80s by howard under the name propositions-as-types and given the name curry-howard isomorphism because haskell curry noticed in the 1930s that the implication is right adjoint to the conjunction. in the type-theoretic treatments, the adjunctions got lost, although every reader of gentzen's thesis may format sequent calculus as a sequence of posetal adjunctions...
lawvere's paper was reissued in TAC at some point. but lawvere himself might be a little guilty of the fact that it all got forgotten, since he tried to say too much using too big words.
lambek had a funny little paper, telling how they understood the world through adjunctions sitting in cafes in zurich cca 1965, i think. "they" being he and lawvere and freyd, i think. the title is something like "the influence of heraclitus on category theory" --- and he explains by examples how adjunctions capture the logical principles of dialectics, as it propagates through western philosophy and science from heraclitus onward. lambek's own theory of syntax (which makes him more perhaps more famous in linguistics than in math) was an obvious derivative of his treatment of modules in terms of adjunctions --- which was basically non-commutative linear logic, just in the late 50s...
for lawvere, the most painful thing would be if adjunctions are thought of as a general pattern of many particular cases. they are a concrete law of nature of logic, which directly drives all of its instances. not an accidental pattern.
so that question on stack exchage would be very painful for them.
David Michael Roberts (Apr 21 2023 at 03:00):Lambek, J. (1981). The Influence of Heraclitus on Modern Mathematics. In: Agassi, J., Cohen, R.S. (eds) Scientific Philosophy Today. Boston Studies in the Philosophy of Science, vol 67. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8462-2_6
Jules Tsukahara (Apr 21 2023 at 05:21):I just discovered Galois connections while reading about Von Neumann's Bicommutant Theorem, but what dusko is saying here seems mind-blowing
John Baez (Apr 21 2023 at 05:42):Thanks, @David Michael Roberts! Lambek was a cool dude. He used to send me his papers about quaternions.
Jean-Baptiste Vienney (Apr 21 2023 at 06:07):Thank you for these references and explanations @dusko. Reading you is a bit like listening to Jean-Yves Girard: you hear very interesting ideas but they are sadly, or amusingly, parasitized by the tone of the speaker who tries to make a show and discredits a bit himself by slightly lacking of nuance at times.
Morgan Rogers (he/him) (Apr 21 2023 at 06:36):My talk at SYCO this morning is about a Galois connection. Maybe I'll come back and talk about it here later.
John Baez (Apr 21 2023 at 19:17):@Jean-Baptiste Vienney - I urge you not to turn the conversation into a discussion of other people's defects. Nobody ever reforms due to such criticisms, so they really just make the atmosphere less pleasant.
John Baez (Apr 21 2023 at 19:18):You are now also likely to discover how Dusko responds to jabs like that.
Fabrizio Genovese (Apr 21 2023 at 19:21):...But you are also lucky because (to my knowledge) Girard is not on this server. I'm pretty sure his response would have been entertaining as well :stuck_out_tongue:
John Baez (Apr 21 2023 at 19:31):I bet!
John Baez (Apr 21 2023 at 19:33):Returning momentarily to math, Simon Willerton made the nice point that both "Galois-theoretic" Galois connections, which involve a group acting a set in a structure-preserving way, and "algebraio-geometric" Galois connections, which involve the vanishing of functions at points, are special cases of an extremely simple pattern.
John Baez (Apr 21 2023 at 19:34):Moreover, this pattern is also fundamental to formal concept analysis, an approach for creating "concepts" given a set of things and a set of predicates on these things.
Jean-Baptiste Vienney (Apr 21 2023 at 20:50):Two years ago, when I was still in Marseille, my supervisor relayed a message to the logic group, on which Girard was, about the "World Logic Day" which was created by a guy working on the subject of paraconsistent logic. And he didn't like this message. Soon after, I received a message on my mailbox, which was the reply of Girard. I've never received any other message like this in my life. He was talking about Trump, a terrorist organisation in Brazil composed of Japanese immigrants during World War II named Shindo Renmei (I had to do some research to understand his few phrases) and other people I didn't know much about. At the end, there was his moto in Latin (invented by him) that required one more time the help of Google (translate this time) to be understood.
The only other day I've been more or less in contact with Girard was at the Linear Logic Winter School in 2022, also in Marseille. It was something like 5 days long but he didn't show up before the very last hour, for the final talk, by him. His talk was the only one in French, but hopefully it was indicated on the first slide that "subtitles in English" will appear on these slides during the talk. It was really interesting, the talk was mainly composed of political and philosophical considerations, with a bit of mathematics at times (when he emphasized that it was "complicated mathematics", naming a complicated topology, something like fidèlement-plat quasi compact topology with a lot of words but related to C*-algebras if I remember, but saying that he doesn't remember which one exactly it is). At the end someone asked what was the definition of logical scientism or something like this. He said a few words, but then suddenly stopped the talk saying that he was "tired of hearing himself". After that, the organizers told him that they can go outside if he doesn't feel well and he was still doing great gestures, wiping his forehead as if he was going to die etc...
One of the most interesting point of the talk was when he said nonchalantly "I've never said that" and gave a little new idea. There was like 50 people in the room and I saw several of them (especially the most experienced people, with tenure, not the students who came to learn linear logic) taking a lot of notes and being trembling during two minutes. I really thought it was ridiculous, my method of doing research being mainly to walk in the parks and I'm always too lazy to even read completely the published papers which must be interesting for me. After the talk, I went to the library and I saw guys writing frenetically on the whiteboard trying to understand the few phrases of Girard containing this original content, I really thought there were ridiculous too.
Since that time, I've read a bit about the history of science and it happens very often that people who do important things are comedians like this, so I'm no more surprised by this kind of comedy, when I see someone like this, I'm just thinking that it is something in nature that creativity his correlated with some special social behaviour in these persons and I'm just laughing. Also, I know that there are probably very important informations hidden in their words if you're able to decode.
And I don't think that Girad will never go on this Zulip. First he hates category theory (most of the time, in some papers there was category theory but it was always written with co-authors) and secondly if you watch some of his pictures, you will understand, that he can't go on a place like this open to everyone.
Jean-Baptiste Vienney (Apr 21 2023 at 21:27):Sorry, I just wanted to share some anecdotes about Girard, now I'm done :sweat_smile: (I have yet another funny one that Phil Scott told me about how his girlfriend made him wrote a magnificent phrase in the paper on Bounded Linear Logic in case)
David Egolf (Apr 21 2023 at 22:02):Morgan Rogers (he/him) said:
My talk at SYCO this morning is about a Galois connection. Maybe I'll come back and talk about it here later.
I enjoyed watching the start of your talk! I appreciated the efforts you made to explain concepts to a wider audience.
It was interesting to hear about the Galois connection you touched on - relating subgroups of the permutations of a set and relations on that set. It reminds me a little of one of the examples John Baez mentions in the "Bargain-Basement Mathematics" article linked above, where we can consider the automorphisms of a field that fix every element of a field and a field that are both inside .
Also, I had not really heard about "model theory" before! It was interesting to look at the start of that book by Hodges that you mentioned.
John Baez (Apr 21 2023 at 22:16):You can take any sort of structure on a set and get a Galois connection between 'subsets having that structure' and 'subgroups of the group of permutations preserving that structure'. Galois was lucky to do an example that let him prove exciting stuff like "you can't solve the general quintic using radicals".
Morgan Rogers (he/him) (Apr 22 2023 at 11:21):David Egolf said:
Morgan Rogers (he/him) said:
My talk at SYCO this morning is about a Galois connection. Maybe I'll come back and talk about it here later.
I enjoyed watching the start of your talk! I appreciated the efforts you made to explain concepts to a wider audience.
It was interesting to hear about the Galois connection you touched on - relating subgroups of the permutations of a set and relations on that set. It reminds me a little of one of the examples John Baez mentions in the "Bargain-Basement Mathematics" article linked above, where we can consider the automorphisms of a field that fix every element of a field and a field that are both inside .
Also, I had not really heard about "model theory" before! It was interesting to look at the start of that book by Hodges that you mentioned.
I'm glad you enjoyed it! The recording is available here, although I'm disappointed about the audio issues, which are a little distracting. The slides are also on the conference website.
As John has been pointing out, constructing Galois connections is often actually pretty easy, and stating the one in my talk didn't require any of the topos theory that I went on to present (incidentally, I introduced toposes from different perspectives, all of which I used in the proof in the later part of the talk, so watch this if you've ever wondered what "doing topos theory" might mean). The depth always seems to come in understanding the fixed points, or more critically in identifying whether the connection you have constructed is a shadow of a deeper connection between the two sides.
I realise now that I never explained the title of the talk! Model theory is the study of set-theoretic semantics of formal theories. The early school of categorical logic comes from asking what we need to extract from sets to still find semantics of these theories. It might be stretching history a little, but the treatment of toposes via their internal logics can be motivated from a desire to expand set theory beyond its traditional axiomatization. Meanwhile, the theory of accessible and presentable categories comes from studying properties of the categories of models of these theories. So there's lots of precedent for taking ideas from model theory and seeing how they can be recovered in a categorical setting in order to subsequently expand upon them.
dusko (Apr 27 2023 at 02:00):Jean-Baptiste Vienney said:
Thank you for these references and explanations @dusko. Reading you is a bit like listening to Jean-Yves Girard: you hear very interesting ideas but they are sadly, or amusingly, parasitized by the tone of the speaker who tries to make a show and discredits a bit himself by slightly lacking of nuance at times.
please say more. i seem to be completely unaware of my tone...
in so far as i have some insight into what is in my head, a comparison with girard must be a misunderstanding. he was kreisel's best student! i was the worst student of anyone who tried to teach me anything. he was a leader of a school. my only consistent property is that if i see a group of people going in some direction, i swim across. (i grew up in a communist country in an anticommunnist house, but my mother and my grandmother explained very early on that in every conflict it usually doesn't take long before all sides are wrong...)
if i lack nuance, it is because i don't see it. pointers will be helpful and appreciated. but no pressure. thanks in any case.
hmm "make a show". what would i do with a show if i managed to make it/ as far as i can see, all choices that i ever made in my life were opposite from making a show. i drove a truck not just through every party membership but through paying careers as soon as they emerged from behind horizon... but there are surely many things that we do without choosing to do them... maybe the whole thing is a show :)))
Jean-Baptiste Vienney (Apr 28 2023 at 19:28):@dusko I'm pleasantly surprised by how you react. That's incredible when you can discuss with somebody who is willing to be refuted but even more who can reconsider himself because of what you said. I think I'm able to accept critics if they are intelligent but a lot of people seem to run away from conflict and so don't like criticize or being criticized. It's unfortunate because lively discussion are more able to give rise to interesting conclusions than completely flat ones.
When I was younger, I read Gorgias by Plato and I felt in love with these phrases of Socrates:
I am one of those who are very willing to be refuted if I say anything which is not true, and very willing to refute any one else who says what is not true, and quite as ready to be refuted as to refute; for I hold that this is the greater gain of the two, just as the gain is greater of being cured of a very great evil than of curing another. For I imagine that there is no evil which a man can endure so great as an erroneous opinion about the matters of which we are speaking; and if you claim to be one of my sort, let us have the discussion out, but if you would rather have done, no matter;–let us make an end of it.
but I've rarely found anybody who seems ready to follow these rules.
I was comparing you with Girard because you share some features but I don't think you're exactly like Girard. He is the only living person I know that I think is a kind of genius. He has created or discovered something new and huge.
You probably lack of nuance when you say
a list of things that are like each other is way less than what lawvere discovered in his 1969 Dialectica paper: that ALL LAWS OF LOGIC ARE ADJUNCTIONS".
Is there a way to find all these adjunctions by a universal method? Lawvere says in his paper:
Needless to say, viewing all these constructions explicitly as adjoint situations seems to have certain formal and conceptual utility apart from any philosophical attempt to unify their necessity.
So he didn't find any magical formula to generate a list of all the interesting adjunctions, and your phrase sounded like if he did. He just mentioned that they can be gathered under a same philosophical idea.
Maybe the philosophical ideal is sufficient as a universal method for you, even if it's not a mathematical one, and probably I should read what remains of Heraclitus to get used to this method and understand the source of all adjunctions... Well, I will do it one day, I'm interested by Heraclitus.
Other instance of lack of nuance:
so that question on stack exchange would be very painful for them
No, I don't think so. There is no mathematical method to find all these adjunctions. As far as I understand, just a philosophical one. In his papers Lawvere gives list of examples of adjunctions as in this question on stack exchange.
all choices that i ever made in my life were opposite from making a show
Each time you write a message on this Zulip it's a show. Are you not aware that you decided to write without using any capital letter either at the beginning of the phrases, for the "I" or for the proper nouns? Clearly one of your goal must be to try to look different than other people and so this is more or less making a show. Secondly, you always try not to follow the group and to contradict everyone, it also makes your comments looks like a show.
(We would have to define more clearly what is making a show to speak more clearly but I know that you think words convey their own meaning and or not just interchangeable packs of letters like Wittgenstein would say (I think this is opposition is very reductive but I have just read a bit of Wittgenstein so I don't understand him very good right now so I can't say).)
You're way of writing messages is the main thing that makes me think to Girard: you kinda make a show, what you say is interesting but you want deeply (or not but you make it) to be in opposition with other people. Creating this opposition can have contradictory effects: either it makes you appear as unique, either it makes you appear as a fool. And like Girard, I know that you're not a fool because you wrote plenty of interesting papers. So hearing someone like this is quite funny.
dusko (Jul 02 2023 at 10:34):Jean-Baptiste Vienney said:
dusko I'm pleasantly surprised by how you react. That's incredible when you can discuss with somebody who is willing to be refuted but even more who can reconsider himself because of what you said. I think I'm able to accept critics if they are intelligent but a lot of people seem to run away from conflict and so don't like criticize or being criticized. It's unfortunate because lively discussion are more able to give rise to interesting conclusions than completely flat ones.
When I was younger, I read Gorgias by Plato and I felt in love with these phrases of Socrates:
I am one of those who are very willing to be refuted if I say anything which is not true, and very willing to refute any one else who says what is not true, and quite as ready to be refuted as to refute; for I hold that this is the greater gain of the two, just as the gain is greater of being cured of a very great evil than of curing another. For I imagine that there is no evil which a man can endure so great as an erroneous opinion about the matters of which we are speaking; and if you claim to be one of my sort, let us have the discussion out, but if you would rather have done, no matter;–let us make an end of it.
but I've rarely found anybody who seems ready to follow these rules.
I was comparing you with Girard because you share some features but I don't think you're exactly like Girard. He is the only living person I know that I think is a kind of genius. He has created or discovered something new and huge.
You probably lack of nuance when you say
a list of things that are like each other is way less than what lawvere discovered in his 1969 Dialectica paper: that ALL LAWS OF LOGIC ARE ADJUNCTIONS".
Is there a way to find all these adjunctions by a universal method? Lawvere says in his paper:
Needless to say, viewing all these constructions explicitly as adjoint situations seems to have certain formal and conceptual utility apart from any philosophical attempt to unify their necessity.
So he didn't find any magical formula to generate a list of all the interesting adjunctions, and your phrase sounded like if he did. He just mentioned that they can be gathered under a same philosophical idea.
Maybe the philosophical ideal is sufficient as a universal method for you, even if it's not a mathematical one, and probably I should read what remains of Heraclitus to get used to this method and understand the source of all adjunctions... Well, I will do it one day, I'm interested by Heraclitus.
Other instance of lack of nuance:
so that question on stack exchange would be very painful for them
No, I don't think so. There is no mathematical method to find all these adjunctions. As far as I understand, just a philosophical one. In his papers Lawvere gives list of examples of adjunctions as in this question on stack exchange.
all choices that i ever made in my life were opposite from making a show
Each time you write a message on this Zulip it's a show. Are you not aware that you decided to write without using any capital letter either at the beginning of the phrases, for the "I" or for the proper nouns? Clearly one of your goal must be to try to look different than other people and so this is more or less making a show. Secondly, you always try not to follow the group and to contradict everyone, it also makes your comments looks like a show.
(We would have to define more clearly what is making a show to speak more clearly but I know that you think words convey their own meaning and or not just interchangeable packs of letters like Wittgenstein would say (I think this is opposition is very reductive but I have just read a bit of Wittgenstein so I don't understand him very good right now so I can't say).)
You're way of writing messages is the main thing that makes me think to Girard: you kinda make a show, what you say is interesting but you want deeply (or not but you make it) to be in opposition with other people. Creating this opposition can have contradictory effects: either it makes you appear as unique, either it makes you appear as a fool. And like Girard, I know that you're not a fool because you wrote plenty of interesting papers. So hearing someone like this is quite funny.
Jean-Baptiste Vienney sorry that i disappeared. i read your comments now, and they are very kind and certainly useful for me. chatlists are a low bandwidth channel, and my lack of nuance is genuinely a lack of voice. i am sorry :)
(to think that i once struggled how to translate Verlaine's "nuances, nuances sans couleurs"... :)))