Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

September 7, 2010

Grothendieck’s “Tohoku” paper

Posted by John Baez

Michael Barr has spent a lot of time and effort translating Grothendieck’s legendary “Tohoku” paper into English. If you don’t mind violating Grothendieck’s stated wishes, you can now read this translation:

It’s about abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence.

In the preface, Michael Barr writes:

We found and fixed (mostly silently) innumerable errors in the text and doubtless introduced many of our own. In a number of cases, we have simplified Grothendieck’s somewhat tortured sentences that sometimes went on interminably with parenthetical inserts. In a few cases, we have updated the language (for example, replacing “functor morphism” by “natural transformation”). In one or two places, we were unable to discern what he meant.

One curiosity is that Grothendieck seems to have had an aversion to the empty set. Products and sums are defined only for non-empty index sets and even finitely generated modules are required to have at least one non-zero generator. The zero module is not considered free (although it is, obviously, finitely generated). Except that his definition of complete is incomplete, this aversion does not really affect anything herein.

Grothendieck treats a category as a class of objects, equipped with a class of morphisms. This differs from both the original view expressed in Eilenberg and Mac Lane and in later and current views, in which a category consists of both the objects and arrows (or even of the arrows alone, since the objects are recoverable). This shows up in several ways, not least that he writes ACA \in C to mean that AA is an object of CC and, most importantly, he says “CC is a set” to say what we would express as “CC is small” or “CC has a set of objects”.

One point to be made is that Grothendieck systematically uses “==” where we would always insist on “\cong”. The structuralists who founded Bourbaki wanted to make the point that isomorphic structures should not be distinguished, but category theorists now recognize the distinction between isomorphism and equality. For example, all of Galois theory is dependent on the automorphism group which is an incoherent notion in the structuralist paradigm. For the most part, we have replaced equality by isomorphism, when it seems appropriate.

These comments would be incomplete without a word about copyright issues. We do not have Grothendieck’s permission to publish this. His literary executor, Jean Malgoire refused to even ask him. What we have heard is that Grothendieck “Does not believe in” copyright and will have nothing to do with it, even to release it. So be it. We post this at our peril and you download it, if you do, at yours. It seems clear that Grothendieck will not object, while he is alive, but he has children who might take a different view of the matter.

Despite these comments, the carrying out of this translation has been an interesting, educational, and enjoyable activity. We welcome comments and corrections and will consider carefully the former and fix the latter.

UPDATE, MARCH, 2010. Since the above was written in Dec. 2008, there has been a new development. Grothendieck has asked that all republication of any of his works (in original or translation) be ended. He has not actually invoked copyright (which, as stated above, he does not believe in), but asked this as some sort of personal privilege. This makes no sense and Grothendieck never expressed such a wish before. I personally believe that Grothendieck’s work, as indeed all mathematics including my own modest contributions, are the property of the human race and not any one person. I do accept copyright but only for a very limited time. Originally in the US, copyright was for seven years, renewable for a second seven. These periods were doubled and then doubled again and the copyright has now been extended essentially indefinitely, without the necessity of the author’s even asking for a copyright or extension. This is a perversion of the original purpose of copyright, which was not to make intellectual achievements a property, but rather to encourage the publication, eventually into the public domain, of creative efforts.

In any case, you should know that if you copy, or even read, this posting you are violating Grothendieck’s stated wishes, for what that is worth.

Posted at September 7, 2010 12:24 PM UTC

TrackBack URL for this Entry:

15 Comments & 1 Trackback

Re: Grothendieck’s “Tohoku” paper

From the introduction
… In a few cases, we have updated the language (for example, replacing “functor morphism” by
“natural transformation”). …

- it seems that “functor morphism” is actually more modern right now, and is better.

Posted by: anonymous on September 7, 2010 6:51 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Care to explain that, anonymous? Why “more modern”, and why “better”?

Posted by: Tom Leinster on September 7, 2010 9:44 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

It is just my opinion, but it seems that to say “morphism between functors” instead of “natural transformation” unifies notions, minimizes terminology. Functors are 1-morphism, natural transformations are 2-morphisms, why do we need another name?

Posted by: anonymous on September 8, 2010 10:33 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

There is also a Russian translation of Tohoku, published decades ago.

Here is a scan:

and the link is mentioned in nlab entry

Posted by: Zoran Skoda on September 7, 2010 7:52 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

What’s wrong? I open this djvu-file with WinDjView, but I can see only one empty page.

Posted by: osman on September 7, 2010 8:17 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Yeah, the djvu link doesn’t work for me either. Usually I have no problems viewing djvu files. It’s only 92375 bytes, which is suspiciously small even for djvu.

Posted by: Tom Leinster on September 7, 2010 9:37 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Before we solve the problem (thank you for alerting me), with the presently corrupted file in nlab you can use the temporary location

Posted by: Zoran Skoda on September 8, 2010 1:33 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

I kind of like Barr’s translation better because of his gloss. This said, having more than one translation is nice.
Thanks for posting the Russian scan.

Posted by: Eugene Lerman on September 8, 2010 3:44 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

One good way to ensure that your important work is copied and recopied until it achieves information immortality is to declare your intention to refuse copyrights and to demand the removal of republications. This is demonstrated by the comments at the Secret Blogging seminar–people were rushing to save personal copies of what previously had been publicly archived papers! Since I suspect this subversive intent, it can’t be wrong to take a look…

Posted by: stefan on September 7, 2010 8:44 PM | Permalink | Reply to this
Read the post Twitter Trackbacks
Tracked: September 8, 2010 10:57 AM

Re: Grothendieck’s “Tohoku” paper

I would like to make it clear that the translation was by my wife, Marcia L. Barr, and me. She is a professional translator and dictated a preliminary translation of each sentence to me and I put it into “mathematicese” and then into TeX.

As for the question of “functor morphism” vs. “natural transformation”, while I do agree the former would have been a more evocative term, I have just not seen it actually used so I can hardly think it is “more modern”.

Posted by: Michael Barr on October 3, 2010 1:49 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Hello. The link for Barr’s translation isn’t working for me.

Posted by: John on November 16, 2011 1:52 AM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Yeah, me neither now. Maybe that ftp site is down.

Posted by: John Baez on November 16, 2011 7:54 AM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Updated link in the answer to this math stack exchange question.

Posted by: S on March 2, 2012 9:35 PM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

To M. Barr:

“… you are violating Grothendieck’s express(ed) wishes, for what that is worth.”

“for what that is worth.” How arrogant. How rude.

Sincerely, Philip Dabous

Posted by: Philip Dabous on March 13, 2014 12:53 AM | Permalink | Reply to this

Re: Grothendieck’s “Tohoku” paper

Via, Math Stackexchange the location of Barr’s English translation of Grothendieck’s “Tohoku” has moved to I’ve also updated the link on the nLab’s Tohoku page.

Posted by: RodMcGuire on November 28, 2014 6:54 PM | Permalink | Reply to this

Post a New Comment