bourbaki

Bourbaki's Éléments de mathématique as Markdown, with permanent tags, translations, and worked exercises.

The source PDFs go in, Markdown comes out. Every numbered statement gets a four-character tag that never changes, so you can cite a result and have the citation still work after the text is re-extracted, re-split, or renumbered. That idea is lifted straight from the Stacks Project.

This repo is the corpus only. The code that builds it lives in tamnd/bourbaki-solver.

What is in scope

All twelve Books of the Éléments, in the English translation where one was printed and in the French original everywhere else. 43 volumes, 14989 pages, of which 15 volumes and 6648 pages are English and 28 volumes and 8341 pages are French.

Book	English	French	Volumes	Pages
Theory of Sets	I to IV	none held	1	418
Algebra	I to VIII	I to X	8	3690
General Topology	I to X	I to X	4	1511
Functions of a Real Variable	I to VII	I to VII	2	683
Topological Vector Spaces	I to V	I to V	2	740
Integration	I to IX	I to IX	7	1732
Commutative Algebra	I to VII	I to X	5	1733
Variétés différentielles et analytiques	never translated	fascicule de résultats	1	190
Lie Groups and Lie Algebras	I to IX	I to IX	8	2182
Théories spectrales	never translated	I to V	2	925
Topologie algébrique	never translated	I to IV	1	512
Elements of the History of Mathematics	whole	whole	2	673

The French is not a fallback for the English. Three Books were never translated, Algèbre chapters IX and X and Algèbre commutative chapters VIII, IX and X exist in French only, and where both printings are held the French is the original a disputed English sentence gets checked against. So both are in scope and both carry the same tags.

Two gaps worth stating plainly. There is no French Théorie des ensembles here, only the English translation of it. And Groupes et algèbres de Lie, Chapitres 7 et 8 is a partial file: 61 pages that open at Chapitre VII and stop in the middle of the exercises around printed page 65, so the English Chapters 7-9 is the only complete copy of that material.

bourbaki books list prints the volume by volume detail, and manifests/books.yaml carries it as data, with the SHA-256 and page count of every file so that a swapped or re-downloaded PDF is caught rather than silently extracted.

The PDFs are not here and will not be. They are copyright Springer and N. Bourbaki. This corpus is for personal study.

Three extraction paths, not one

What a volume costs to read depends entirely on what its own text layer is worth, and the library splits three ways. bourbaki books add measures this rather than guessing it, by sampling the images and the text of a band of body pages, and records the answer as text_layer in the manifest.

Text layer	Volumes	What it means
native	6	Born digital. pdftotext -layout gives real text and real mathematics.
ocr	34	A scan somebody has already run OCR over. Good enough to read a running head off, useless for mathematics.
none	3	A scan with no text at all. Even the page map has to come out of vision OCR.

The six native volumes are Algebra, Chapter 8 and Lie Groups and Lie Algebras, Chapters 7-9 in English, and Algèbre chapitre 8, Théories spectrales chapitres 1 et 2, Théories spectrales chapitres 3 à 5 and Topologie algébrique chapitres 1 à 4 in French. They are cheap and they go first. The three with no text at all are Commutative Algebra at 642 pages, General Topology Chapters 5-10 at 372 pages and Algèbre chapitre 10 at 222 pages, and they are the most expensive volumes in the library.

The 34 in the middle are the ordinary case. Their OCR layer renders a pair of braces in Theory of Sets as R! x, y I, so nothing in it can be trusted as mathematics, but it is legible enough to build the page map from before a single page goes to vision OCR, which saves the expensive pass on all 34.

Running heads differ per volume

Worth knowing before you touch the page-map code, because each volume prints its locators differently:

1998, Ch. I     I                    ALGEBRAIC STRUCTURES
                                     p-GROUPS                         §6.5
2003, Ch. IV    A. IV. 2             POLYNOMIALS AND RATIONAL FRACTIONS   §1
                No. 2                POLYNOMIALS                      A.IV.3
2023, Ch. VIII  No 4                 POLYNOMIALS WITH ...              A VIII.13

Bourbaki cross-references are page based, as in VIII, p. 3, Proposition 3, so the page map is not decoration. It is the key the reference resolver joins on. The 1998 volume does not print a page locator in its running head at all, which is why its page map is anchored and interpolated rather than read off directly.

Tags

#### Proposition 6 {#alg-viii-s1-prop-6 .statement tag=0A3F}

tags/tags is append only, one line per statement:

0A3F,alg-viii-s1-prop-6

A tag is never reused and never edited. If a label has to change, the tag follows the statement and the old label goes to tags/aliases. Every edition uses the same tag for the same statement, the French original included, so a tag is the one identifier that works across all of them and is what lets a translation be checked against the French rather than only against the English it was made from.

tags/runs says where one assignment stopped and the next began, one line per merge. Tags are handed out in reading order, so the tags of a file climb on the run that assigned them, and they do not climb across runs: a statement added to the middle of a § a month later takes a tag above everything under it. The boundaries are what let the audit tell that correct edit from a heading somebody pasted the wrong tag on to.

Layout

content/en/<book>/<CH>/NN_sN_<slug>.md         one file per §
content/en/<book>/<CH>/exercises/sN/NN.md      one file per exercise
content/fr/...                                 same tree, extracted from the French printing
content/{vi,zh,ja}/...                         same tree, translated
content/solutions/<lang>/<book>/<CH>/sN/NN.md  verified solutions
tags/                                          permanent tag index
manifests/                                     books, TOC, page maps, refs, glossary, errata, editions
figures/                                       cropped diagrams, small, committed
reports/                                       audit, usage, coverage, scorecards
imports/<book>/chapter_<n>/<n>.<m>.md          read off share links, not yet checked

content/fr is not a translation and is never generated from content/en. It is what the French volume prints, extracted the same way the English is, which is why it carries no translated_from and is exempt from the translation rules.

Because the two printings sit side by side, the places they disagree get noticed, and there are two manifests for what is found. manifests/errata.yaml is for a printing that is wrong, and it carries the words the page has and the words to read instead. manifests/editions.yaml is for the other case, where the printings differ and both are right: chapter VIII § 2 has twenty exercises in the 2023 English and nineteen in the 2012 French, the nineteen are the same nineteen, and the twentieth is one the later printing added. A difference like that looks exactly like a page nobody read, so the reading that settles it is written down once instead of being done again by whoever counts next.

pdf/, images/ and work/ are gitignored. Nothing large or copyrighted is committed.

imports/ is deliberately outside content/. It holds sections read off public ChatGPT share links, which cost an HTTP GET rather than 150 seconds of a browser per page, and which nobody has yet held against the printed book. The audit runs over content/, so anything dropped in there would pass all 66 rules by default. See imports/README.md.

Coverage

Book	Chapter	Sections	Statements	Exercises	Tagged	Pages
Algebra	I	0 of 10	0	0	0	3
Algebra	II	0 of 12	0	0	0	1
Algebra	III	0 of 12	0	0	0	1
Algebra	IV	0 of 6	0	0	0	0
Algebra	V	0 of 17	0	0	0	0
Algebra	VI	0 of 2	0	0	0	1
Algebra	VII	0 of 5	0	0	0	1
Algebra	VIII	25 of 25	710	317	1027	964
Theory of Sets	I	6 of 6	16	35	51	50
Theory of Sets	II	6 of 6	109	41	150	65
Theory of Sets	III	7 of 7	222	120	342	127
Theory of Sets	IV	3 of 3	28	15	43	147
Topological Vector Spaces	I	0 of 3	0	0	0	0
Topological Vector Spaces	II	0 of 8	0	0	0	0
Topological Vector Spaces	III	0 of 6	0	0	0	0
Topological Vector Spaces	IV	0 of 6	0	0	0	0
Topological Vector Spaces	V	0 of 4	0	0	0	0
Functions of a Real Variable	I	0 of 4	0	0	0	2
Functions of a Real Variable	II	0 of 3	0	0	0	0
Functions of a Real Variable	III	0 of 2	0	0	0	1
Functions of a Real Variable	IV	0 of 2	0	0	0	1
Functions of a Real Variable	V	0 of 5	0	0	0	1
Functions of a Real Variable	VI	0 of 3	0	0	0	2
Functions of a Real Variable	VII	0 of 2	0	0	0	2
Integration	I	0 of 1	0	0	0	0
Integration	II	0 of 2	0	0	0	0
Integration	III	0 of 4	0	0	0	2
Integration	IV	0 of 7	0	0	0	0
Integration	V	0 of 8	0	0	0	0
Integration	VI	0 of 4	0	0	0	0
Integration	VII	0 of 5	0	0	0	0
Integration	VIII	0 of 5	0	0	0	0
Integration	IX	0 of 7	0	0	0	0
Lie Groups and Lie Algebras	IV	0 of 3	0	0	0	1
Lie Groups and Lie Algebras	V	0 of 7	0	0	0	3
Lie Groups and Lie Algebras	VI	0 of 4	0	0	0	7
Lie Groups and Lie Algebras	VII	7 of 7	139	66	205	66
Lie Groups and Lie Algebras	VIII	13 of 13	299	172	471	211
Lie Groups and Lie Algebras	IX	11 of 11	244	112	356	152
Topologie algébrique	I	0 of 6	0	0	0	150
Topologie algébrique	II	0 of 5	0	0	0	78
Topologie algébrique	III	0 of 5	0	0	0	110
Topologie algébrique	IV	0 of 6	0	0	0	158
General Topology	I	0 of 11	0	0	0	0
General Topology	II	0 of 4	0	0	0	0
General Topology	III	0 of 7	0	0	0	0
General Topology	IV	0 of 8	0	0	0	0
Théories spectrales	I	0 of 8	0	0	0	197
Théories spectrales	II	0 of 3	0	0	0	136
Théories spectrales	III	0 of 6	0	0	0	143
Théories spectrales	IV	0 of 5	0	0	0	228
Théories spectrales	V	0 of 4	0	0	0	194

78 of 325 sections are in the corpus, 24 per cent. 1767 statements and 878 exercises, 2645 of them carrying a permanent tag.

The table is one row per chapter of the volumes that have a table of contents. 28 further volumes and 8211 pages are registered in manifests/books.yaml with no table of contents read off them yet, so none of their chapters are counted above.

Building it

Registering a volume probes it and writes what it measured, so this is run once per file and then the manifest is the record:

export BOURBAKI_CORPUS=$PWD
bourbaki books add "pdf/en/Algebra Chapter 8 (2023, Springer Nature).pdf" \
  --id alg-viii --book alg --lang en --chapters VIII
bourbaki books add "pdf/fr/Algèbre_ Chapitre 8 (2012, Springer).pdf" \
  --id alg-viii-fr --book alg --lang fr --chapters VIII
bourbaki books list
bourbaki books verify

Then per volume. A native volume is three commands and a few minutes:

bourbaki pagemap build --book alg-viii
bourbaki extract       --book alg-viii
bourbaki assemble      --book alg-viii

A scan is four, and the middle two are the ones that take days:

bourbaki pagemap build --book alg-i-iii
bourbaki render        --book alg-i-iii --dpi 300
bourbaki ocr           --book alg-i-iii
bourbaki assemble      --book alg-i-iii

A volume with no text layer at all cannot have its page map read off the file, so on those three pagemap build runs after render and ocr rather than before. That is the third path and it is why Commutative Algebra, General Topology Chapters 5-10 and Algèbre chapitre 10 are the last volumes anybody should pick up.

The rest is corpus wide and runs after any volume changes:

bourbaki tags assign && bourbaki tags merge && bourbaki tags verify
bourbaki refs build
bourbaki report coverage --write-readme
bourbaki audit --report reports/audit.md

assemble writes the section files, the exercise files and the two manifests in one pass, and sweeps whatever an earlier split left behind. assemble --check is the same run with the writing taken out and a diff put in its place, which is what CI runs.

OCR runs against a small fleet of hosts over SSH. Round trips are slow, roughly 150 seconds a call, so every long stage is resumable and safe to interrupt.

Where the numbers are

reports/ is generated and checked by CI, so it is the state of the corpus rather than a snapshot somebody remembered to update. audit.md lists every finding by file and line. extraction-quality.md says how much of each volume has been read and how much of that passes the rules. refs-unresolved.md and out-of-corpus.md are the references that resolve to nothing and the ones that point outside the corpus.

what-it-cost.md and reproduction.md are the two written by hand. The first says what the transport cost, which checks caught what, and what is still wrong, including the numbers that do not flatter the project. The second is the record of cloning both repositories fresh and running the pipeline from the PDFs to see whether it produces what is committed: 2775 of 2868 pages of the six deterministic volumes came back byte for byte, and everything derived from the pages came back identical.

Licence

Transcriptions, translations and solutions here are derived from copyrighted material and exist for personal study only. Éléments de mathématique is copyright N. Bourbaki and its publishers. No source PDF is distributed.