Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Grothendieck, before he disappeared, was working on a manuscript called "Les Derivateurs", which detailed the theory of derivators. Prof. Cisinski has done work with them as he mentioned in this post. However, most of his work is in French, and I was wondering if there are any typed up references in english (the nLab has written notes from a seminar, but that's it). I intend to read the references in French later, but could someone explain or give a reference that explains in English the definition of a derivator and the motivation for them?

share|improve this question
1  
According to Wikipedia the original idea was developed in Pursuing Stacks which is in English as far as I know. –  Steven Gubkin Mar 8 '10 at 14:44
add comment

6 Answers

up vote 13 down vote accepted

For a few references in English, there are the papers of Heller, the main one being:

A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (383) (1988)

There is also a paper I wrote with A. Neeman, in which there is a little introduction to derivators in the second half of: Additivity for derivator K-theory, Adv. Math. 217 (2008), no. 4, 1381-1475

One can see derivators in action in the work of G. Tabuada (he explains Bousfield localization and stabilization in this setting, and compares with the model category point of view): Higher K-theory via universal invariants, Duke Math. J. 145 (2008), no. 1, 121–206 (availabe as arXiv:0706.2420).

share|improve this answer
add comment

George Maltsiniotis has given a mini-course in Seville last year, they include some handwritten notes in English, which are the part 3 of the course:

Lecture_III_Derivators.pdf

For the other parts of the course notes see the conference page.

share|improve this answer
add comment

Some very brief remarks on derivateurs are in Bertrand Toen's habilitation thesis on pages 12-14 and 27-29, available here. They mainly emphasize how derivateurs are sort of a 2-truncated homotopy theory of homotopy theories - nothing to learn derivateur-language from, but maybe a nice addition to the general picture.

share|improve this answer
add comment

My recommendation would be Moritz Groth's excellent notes

He also has a follow-up paper on

as well as a

These are great introductions to all of these topics.

share|improve this answer
add comment

Urs Schreiber updated the nLab page 40 minutes or so ago with an explanation from Prof. Cisinski.

share|improve this answer
add comment

A theory of a very similar flavour can be found in a preprint of Jens Franke.

share|improve this answer
1  
Dead link is dead. :( –  darij grinberg Sep 9 '11 at 13:13
1  
citeseerx.ist.psu.edu/viewdoc/… should work. Notice that this is only part 1 of the preprint while part 2 seems to contain a gap, which is partially filled by a recent preprint of Irakli Patchkoria. –  Lennart Meier Sep 10 '11 at 8:43
    
Irakli Patchkoria's preprint: arxiv.org/abs/1108.6309 –  Rasmus Bentmann Feb 7 '13 at 16:16
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.