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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?

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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
up vote 15 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).

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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:


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

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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.

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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.

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

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

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Dead link is dead. :( – darij grinberg Sep 9 '11 at 13:13
1… 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: – Rasmus Bentmann Feb 7 '13 at 16:16

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