### 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:

- Alexandre Grothendieck, On certain aspects of homological algebra, trans. Michael Barr.

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 $A \in C$ to mean that $A$ is an object of $C$ and, most importantly, he says “$C$ is a set” to say what we would express as “$C$ is small” or “$C$ 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.

## 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.