Les Dérivateurs (The Derivators)

Text of Alexander Grothendieck

Edited par M. Künzer, J. Malgoire, G. Maltsiniotis


"Les Dérivateurs" is a text written by Alexander Grothendieck between October 1990 and the middle of 1991. This manuscript, 1976 pages long, is devoted to the foundations of homotopy theory. The main theme is the notion of derivator, but several chapters are devoted to the homotopy theory of the category of small categories or to the exploration of different variants of model categories. In the enormous chapter next-to-last, the foundations of the theory of accessible categories are studied. The term and the notion of derivator appear for the first time in the section 69 of "Pursuing Stacks". A letter of Grothendieck to Thomason, dating April 2, 1991, is an excellent introduction to the theory of derivators.

This electronic publication of the "Dérivateurs" would have never appeared without the tremendous task of deciphering and typing accomplished by Matthias Künzer and his precious help for the editing. I would like to express my gratitude to him. I would like to thank Jean Malgoire for giving me a copy of the manuscript that Grothendieck gave him in the middle of the nineties and for spending countless hours with me to compare the original with the TeXed version.

The transcription of the manuscript is as faithful as possible. For some minor corrections, or comments of the editors, as well as for the original numbering of the pages, a typewriter font is used within brackets. A question mark in brackets means that we are not sure about the preceding word. The table of contents was established by the editors. Links should point to the (French) main page of derivators, not directly to the ps or pdf files, whose names will change in successive versions. For any remark, comment or correction, please write to

    Georges Maltsiniotis

Some texts related to derivators

Some other pages dedicated to derivators

The 2001 Seminar on Derivators

Pages dedicated to the mathematical texts of Alexander Grothendieck

G. Maltsiniotis , 17/01/2022