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JR Learnstomath (Jan 15 2026 at 09:03):I'm curious about a topic and I'm having a hard time finding more about it, because the original source is in French that includes old words. It is specifically this sentence that I'd like to learn more about the following:
"des immersions d'une surface à bord non vide [un disque] ...je me suis intéressé surtout à une vision dynamique des configurations possibles, avec le passage de l'une à l'autre par déformations continues, qui peuvent se décomposer en composées de deux types d'opérations élémentaires et leurs inverses, à savoir le 'balayage' d'une branche de courbe par dessus un point double, et l'effacement ou la création d'un bigône." (Section 9 of Esquisse d'un programme, Grothendieck. https://en.wikipedia.org/wiki/Esquisse_d'un_Programme (My emphasis on what I think the topics are.)
Specifically, I would like to know more about this "sweeping of a branch of a curve over a double point" idea (my translation, is that right?).
In the English translation, "immersion" is "immersion" and "bigon" is "bigon". I have tried searching for "immersion of disks", "anvil", and plenty of others + "Grothendieck", but I keep getting things about disk drives in water, or very general things on Grothendieck (which is a deluge!).
My questions are just to confirm my guesses:
PS - This kind of "learner's gluing problem" (i.e. How do I "glue" old/original writing to-updated-resources? What are the "software updates" of the notion?) is typical for me as a "self-directed" learner.
PPS - Here's how I got to "anvil":
bigône --update--> bigorne --unpack--> enclume à deux points --translate--> 2-sided anvil
functions are:
update: old French to now French https://projetbabel.org/vosgien/lexiquegerardmer/gerardmerB.htm
unpack: get the definition
translate: now French to now English
@Jean-Baptiste Vienney @Morgan Rogers (he/him) , might you have some guidance here?
Rémy Tuyéras (Jan 15 2026 at 09:35):This is what I think is going on, but I might be missing some details:
https://share.google/images/BYlslHe7nqufA2BFp
(see the image)
the two points coincide and you use some sort of convolution or commutator on the paths...
... from experience, I know that this is the type of operations you would typically see appear if you try to show the 2 out of 3 property for topological spaces
Morgan Rogers (he/him) (Jan 15 2026 at 11:34):Grothendieck wasn't alive so long ago that you needed to translate from "old French" ;)
Morgan Rogers (he/him) (Jan 15 2026 at 11:34):@Rémy Tuyéras 's animation is exactly what I would imagine.
Damiano Mazza (Jan 15 2026 at 13:43):JR Learnstomath said:
- Did "bigône" get translated to "bigon", and is this kind of paper on "bigon + homotopy + theory" what updates "un bigône" https://arxiv.org › pdf › 1101.5658.pdf?
Yes, I would say that a "bigône" is a bigon. Wikipedia says that it may also be called a digon or a 2-gon. It is in the same family as pentagon, hexagon, etc.
This is because a polygon in French is a "polygone". So "bigône" has nothing do with an anvil and is very much contemporary French, albeit admittedly technical and very rarely used.
Maybe today (but perhaps already in the 80s) category theorists would call a bigon a "1-globe"?
John Baez (Jan 15 2026 at 23:37):I think we should let bigons be bigons.
JR Learnstomath (Jan 16 2026 at 08:00):John Baez said:
I think we should let bigons be bigons.
I challenge someone to use this as a section title in a paper #bigonchallenge
JR Learnstomath (Jan 16 2026 at 08:05):Thanks so much! I now know what to explore!
Morgan Rogers (he/him) (Jan 16 2026 at 08:48):Grothendieck's writing certainly is big on challenge, good luck with your studies @JR Learnstomath :)
JR Learnstomath (Jan 16 2026 at 09:17):Morgan Rogers (he/him) said:
Grothendieck's writing certainly is big on challenge, good luck with your studies JR Learnstomath :)
I have to say that in this context, I really appreciated @fosco 's original (co)ends book title!!!
fosco (Jan 16 2026 at 13:51):Morgan Rogers (he/him) said:
Grothendieck wasn't alive so long ago that you needed to translate from "old French" :wink:
https://www.youtube.com/watch?v=pwODwwgE6rA
fosco (Jan 16 2026 at 13:54):JR Learnstomath said:
Morgan Rogers (he/him) said:
Grothendieck's writing certainly is big on challenge, good luck with your studies JR Learnstomath :slight_smile:
I have to say that in this context, I really appreciated fosco 's original (co)ends book title!!!
What context? :smile: The only thing I have that is better than Grothendieck is hair on my head...
JR Learnstomath (Jan 16 2026 at 17:12):Ha! In the context of puns as titles, the #bigonChallenge to use "let
bigons be bigons" as a paper section title, and Morgan saying that M.
Grothendieck is big on challenges. Do I remember well that an early title
of your book was "This is the (co)end my only (co)friend"? It made me think
of song lyrics.
fosco (Jan 16 2026 at 17:15):Ah, I understand now! Yes, it was the perfect title.. But then Cambridge press had to ruin the game :pensive: damn grownups...
JR Learnstomath (Jan 16 2026 at 17:37)::joy: At least you could get away with saying "co/end fu" somewhere, was it in
a description? When I grow up mathematically, I would love to study your
book properly :book: