# Alexander Grothendieck’s letter on Derivators

*02/04/1991*

#### Translator’s note

*This page is a translation into English of the following:*

Grothendieck, A. “Lettre d’Alexander Grothendieck sur les Dérivateurs, 02.04.91.” Edited by Matthias Künzer. [PDF]

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `94f6dce`

*Les Aumettes, 2nd of April 1991.*

Dear Thomason,

Thank you for your letter, and forgive me for having taken so much time to reply.
One reason for this is that, since for the past maybe two months I have been busy thinking about something that started as a small diversion, which I thought I would be able to sort out in a few days (a familiar sentence…), and I delayed this letter week after week.
This thought is not really about homotopical algebra per se, but more about the foundations of category theory, and I have done a lot more than what I need right now [9, Chapter XVIII].
But up until now, I have been convinced that a homotopical algebra (or, in a wider vision, a “topological algebra”) such as I envisage cannot be developed with all the breadth that it has without the aforementioned categorical foundations.
It concerns a theory of (large) categories that I have been calling “*accessible*,” and accessible subsets of these, completely rewriting the provisional theory that I present in SGA 4, §I.9 [2].
I have woven a tapestry of nearly two hundred pages on this seemingly trivial theme, and it would please me to present to you the outlines, if that would be of interest to you.
There are also some intriguing problems that remain, that I feel are difficult, maybe even deep, and that could maybe (who knows) inspire you, or somebody else interested in the foundations of the tool that is category theory.
But all this seems to me part of the domain of the tool, and I would prefer, in this letter, to speak about more central things.
The main ideas were, for the most part, born over 25 years ago, and I see the enduring seedlings of them in my solitary reflections from the years ’56 and ’57, when I first had the need of categories of “coefficients” that were less prohibitively large than the interminable complexes of chains or of cochains, and the idea (after long periods of perplexity) of constructing such categories via passing to a category of fractions (a notion that had to be invented by considering concrete elements) by “inverting” the quasi-isomorphisms.
The principal conceptual work that remained to be done — and that now seems to me as equally fascinating (as much as for its beauty as for its evident impact on the foundations of a cohomological algebra in the spirit of a theory of cohomological coefficients) as in the times of my first loves with cohomology — was to clarify the intrinsic structure of these categories.
The fact that this work, that I had confided to Verdier around 1960, and that was meant to be the subject of his thesis [13], had still not yet been done, even in the case of ordinary and abelian derived categories, which nevertheless have ended up (by necessity) being used daily in both geometry and analysis, says a lot about the state of mentality in the mathematical community with regards to foundations.

This mood of contempt for essential foundational work (and, more generally, for anything that does not follow the fashion of the moment), I mentioned in my last letter, and I also come back to it many times in the pages of *Récoltes et Semailles* [8], and it is one thing (amongst many others) that quite is simply beyond me.
Your response to my letter shows that you absolutely did not understand.
It wasn’t a letter to “complain” about this, or to say that it bothered me.
But it was an impossible attempt to share a pain.
I knew deep down that it was hopeless;
for everybody shuns pain, that is, shuns knowledge (for there is no knowledge from the soul that is free from pain).
A very rare attempt, possibly the only one in my life (at least, I can’t remember another), and probably the last…

There are two directions of ideas, intimately linked, that I want to talk to you about, and that I have been especially keen to develop since the end of October (when I resumed mathematical reflection [9] for an indefinite period).
They are already sketched out here and there (along with a number of other key ideas of topological algebra) in *Pursuing Stacks* [6, Section 69].
In this 1983 reflection, which has helped me a lot these days, I end up spreading myself rather thin by following lateral paths, rather than returning to the essential ideas of my initial point.
As another useful source for anybody interested in these basic questions, I point to two or three letters to Larry Breen, which I thought that I would include in the published text of *Pursuing Stacks* [7].
On the one hand, I would like to talk to you about categories of models and the notion of “derivators” (replacing the defunct “derived categories” of Verdier, which are decidedly inadequate for our needs).
On the other hand, I have a lot to say about

# 1 The only essential structure of a model category is the data of the “localiser” W\subset\operatorname{Mor}({\mathcal{M}})

I also call a category

By their richness of delicately intertwined structures, it is the *closed Quillen triples* *category with cofibrations*, or *with fibrations*, by K.S. Brown [3] (with the quite obvious generalisation given by Anderson [1]) that appears the most beautiful to me.
However, I did not manage to understand the justification for the system of axioms that you proposed to me in your first letter, which placed you somewhere halfway between Quillen [10] and Brown [3].
Your axioms^{1} (seen as extending those of Quillen) seem to me to be prohibitively demanding, when compared to those of Brown–Anderson — apart from the fact, only, that you do not seem to require for the sets

An example: if a structure of fibrations (in the sense of Brown–Anderson) satisfies the familiar condition of “properness” (which is the case for the structures considered first of all by Brown, where all the objects are fibrant over the final object), then we can replace this structure * W-cofibrations*.
In this way, I tend to mostly think of a proper (and not “canonical”) structure of fibrations

*certain*

*not*in the nature of extension-lifting properties of morphisms). This should allow us (by considering infinite “paths”) to construct, in

# 2 Pre-derivators, derivators

When I said that the essential homotopical structures are already contained in the localiser *a priori* that he expresses for every research into foundations that went beyond what he had just done, with such success…)
But the thing becomes evident in the derivators point of view.

The main idea of derivators appeared to me during SGA 5 [5], when it turned out (as discovered by Ferrand [4], who was tasked with editing one of my exposés on traces in cohomology) that Verdier’s notion of derived category did *not* lend itself to the formalism of traces: the trace is not additive for “exact triangles,” since this notion of triangle (thought of as a diagram in the original category) is not fine enough.
To make things work, one has to take the category of morphisms of complexes (for which we have a functorial construction of *mapping cones*), and passed to the derived category of this.
This category maps to that Verdier’s category of triangles by an essentially surjective functor, but which is not at all faithful, let alone fully faithful.
This is the “*original sin*” in the first approach to derived categories attempted by Verdier [12,13] — an approach which in any case, from lack of experience, we could not have done without.
It was then that I was shocked by this fact, seemingly insignificant, that every time we construct a derived category of an abelian category with the help of a category of complexes, this derived category, in a sense, “never comes alone.”
Indeed, for every index category *derivator*”) as a supplementary structure on this category — which continues, however, in the formalism of derivators, to play an important role, under the name of “*base category*” of the derivator.
The same idea had the appearance of working for the non-commutative variants of the notion of derived category, and the work of Quillen appeared to me as a strong incentive to develop this point of view.
But it was only a few months ago that I permitted myself the leisure of verifying that my intuition was good and well justified.
(Stewardship work, almost, as I have done hundreds and thousands of times!)

With this point in mind, it is now very clear that the notion of derivator (even more than that of a model category, which is, in my eyes, a simple “non-intrinsic” intermediary to constructing derivators) is one of the four of five most fundamental notions in algebraic topology, which for thirty years now has been waiting to be developed.
As for notions of comparable scope, I can only think of that of *topos*, and those of * n-categories* and

*on a topos (notions that still haven’t been defined to this day, except for*n -stacks

I now come to the definition, as I understand it, of a “*prederivator*” *domain*” for those that we will consider, i.e. a full subcategory *of objects of a geometric and spatial nature*, like “spaces,” strictly speaking, much more than of an algebraic nature;
just as commutative rings (via their spectra) and the schemes that we construct with them are, for me, geometric-topological objects in essence, and by no means algebraic.
(Algebra being only an intermediary to obtain the geometric vision, which is the fundamental one).
A more or less diametrically opposite case is that where

On the subject of domain, I want to again add that, in my eyes, the domain *depends only, up to equivalence, on the topos defined by X*, and thus only on the category

*covariant*laws

As for the axioms for derivators, the most essential of all is the existence, for every arrow *exact sequence of suspension*, it is the existence of ^{2}
But I point out right away that certain properties of derivators seem important to me, and even when their *statement* relies only on one of the two structures (left or right), they are proven using the existence of the two covariances simultaneously.

To finish these generalities on the notion of derivators, I would like to stress that it is essential, in the notion of prederivator (which is the unique base data), that *general nonsense*”), the data of a *this*, and only this, type of structure that discerns the essential aspects, i.e. intrinsic (independent of the particular chosen model category, for computational purposes, to describe the derivator) to the homologico-homotopical formalism;
which is in its ultimate essence (if I am not very wrong) a *formalism of variance of “coefficients”*.
Just as classical Poincaré duality led me towards the formalism of six operations (or “variances”) that holds in the context of topological spaces as well as that of schemes or analytic spaces (and indeed in others still, such as

To stimulate the usual intuition of familiar homological or cohomological contexts, I have found notations of the following type useful, for a given derivator *objects of the homology and cohomology of X, with coefficients in \xi*”) the objects

*homology*and

*cohomology*(respectively)

*relative to*X over Y , with coefficients in

# 3 The prederivator defined by a model category, and the problem of existence of f_! and f_*

The majority of derivators that I know are defined by means of a model category *all* the axioms) provided only that

On this subject, a remark on the subject of the functoriality of the prederivator associated to a model category *quasi-isomorphisms*” (*with respect to the localisers W and W'*).
It is clear that the quasi-isomorphisms are sent to isomorphisms by the above

*equivalent*as

^{3}There is no downside to working with these “super-structures,” and quite often we cannot do without them. It is, however, important, for a deep understanding, to not lose sight of the essential geometric objects (vector spaces, groups, varieties, derivators) and their intrinsic character, or to let them become blurred.

I do not know if it is reasonable to expect, for the *derivators*, not only prederivators.
In any case, this is one of the questions that we should indeed consider on day (in this world, if there is time, and if not, in another…)^{4}

But I have to return to the case where we are given a fixed model category

It is the moment to say that the work of Anderson [1] (of which you sent me a photocopy), where he claims to give a sketch of a very general theorem of this type (which would have indeed answered my wishes!) is completely a joke — even in the case of a finite ordered set

I am amazed that ten years have passed since this article without anybody apparently noticing that it does not hold water. Obviously, this theorem was like a piece in a museum, a feat for nothing — nobody gave a damn. Even amongst those more “popular” than him, theorems are often no longer an open door to something that we had not seen before and which we see now, nor even a tool for forcing open doors that cannot be opened smoothly — but a trophy. It then doesn’t matter whether it is true or false — it makes absolutely no different any more…

Unless you need clarifications, I think it is useless to enter into details — you are very capable of finding where it screws up by yourself (around any “it is evident that…”). And also, it is time that I stop, even though I have not yet arrived at that which, technically, was planned to be the main substance of my letter: the “factorisation theorem,” and its application to stability theorems for Quillen structures. This will probably be my next letter, if you are interested in continuing this correspondence. In which case I will be very happy to have you as an interlocutor of my cogitations!

In the meantime, best wishes.

# Bibliography

*Bull. Amer. Math. Soc.*

**84**(1978), 765–788.

*Théorie des topos et cohomologie étale des schémas (SGA 4)*. Springer-Verlag, 1972.

*Lecture Notes in Mathematics*.

**269, 270, 305**.

*Trans. Amer. Math. Soc.*

**186**(1974), 419–458.

*On the non additivity of the trace in derived categories*. 2005. http://arxiv.org/abs/math/0506589.

*Cohomologie*\ell -adique et fonctions L (SGA 5). Springer-Verlag, 1977.

*Lecture Notes in Mathematics*.

**589**.

*Pursuing stacks*. 1983.

*Pursuing stacks et correspondance*. 1983.

*Récoltes et Semailles, Témoignage sur un passé de mathématicien*. Montpellier, 1986.

*Les dérivateurs*. 1990. https://webusers.imj-prg.fr/ georges.maltsiniotis/groth/Derivateurs.html.

*Homotopical algebra*. Springer-Verlag, 1979.

*Lecture Notes in Mathematics*.

**43**.

*J. Pure Appl. Algebra*.

**213**(2009), 1916–1935.

*Catégories derivées. Quelques résultats (État 0)*. 1963.

*Des catégories derivées des catégories abéliennes*. Soc. Math. France, 1996.

**239**.

*Bull. Amer. Math. Soc.*

**34**(1997), 1–13.

*Sel. Math., New Ser.*

**7**(2001), 533–564.

*[Editor] The letter from Thomason to Grothendieck on the 3rd of January, 1991 [7]. See also [14,15].*↩︎*[Editor] In “Les Dérivateurs” [9], Grothendieck makes the opposite choice concerning the terminology of left or right derivator, justified by the left or right exactness properties of the prederivator, implied by the existence of*↩︎f_* orf_! , respectively.One of the first comparisons that occurred to me (which was lost en route) seems more striking: the relation between a model category and its associated derivator resembles, to me, that between a complex in an abelian category and the corresponding object in the derived category. And the conceptual effort that I had to make to arrive at the notion of derived category is somewhat similar to that (more modest, in my opinion) which people will have to one day provide in order to access the notion of derivator an the “yoga of derivators” — which can only be acquired by working with them!↩︎