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D_X\colon Y\mapsto D(X\times Y), to which we can apply the general results. The axioms of derivators are such that they are stable under passing from a derivator to an induced derivator.)This suggests to disentangle the notions of “left derivator” (postulating the existence of homological direct images f_!, excluding the cohomological direct images f_*) and “right derivator,” including the dual aspect of the homotopical formalism.2But 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 D\colon\mathsf{Diag}\to\mathsf{CAT} is indeed a 2-functor, and not just a functor;in other words, one has to give not only the D(X) for X\in\mathsf{Diag}, and the f^*=f_D^*=D(f) for the arrows f\colon X\to Y, but for an arrow u\colon f\to f' between two arrows f,f'\colon X\to Y, one has to give a functorial homomorphism u^*\colon {f'}^*\to f^*(with a subscript D if there is any risk of confusion).It must be seen that, conceptually very simple and obvious (and probably for this very reason misunderstood by the “serious mathematician” as “general nonsense”), the data of a 2-functor between 2-categories is a very delicate type of structure, of an apparently new type in maths;and whether or not one wants, it is indeed 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 \mathsf{Cat}, as I am now persuaded), a formalism which, in my opinion (and

Toposes, i.e. to a functor (which we again denote by D) D\colon\underline{\mathsf{Top}}^\circ\to\mathsf{CAT}.I believe that this is always possible, and in an essentially unique way.This is true in all the concrete cases that I have looked at.If, for example, D is the (abelian) derivator defined by an abelian category via the category of complexes and the notion of quasi-isomorphism, we obtain, for every topos {\mathcal{X}} (supposing that the category is that of k-modules, where k is an arbitrary ring), the derived category D({\mathcal{X}},k) of k-modules on {\mathcal{X}}, and this D({\mathcal{X}},k) does indeed depend contravariantly on {\mathcal{X}}.It is true that, when trying to define the covariant laws f_! and f_*, or, more precisely, when trying to establish their existence, we struggle with f_!, with this f_! existing only by placing draconian hypotheses on f.(In any case, I am cheating a bit here, for lacking of having spelled out the restrictions on the degrees of the complexes, like D^+({\mathcal{X}},k) or D^-({\mathcal{X}},k). But here is not the place to enter into technicalities.)As for the axioms for derivators, the most essential of all is the existence, for every arrow f\colon X\to Y in \mathsf{Diag}, of functors f_! and f_* from D(X) to D(Y), left and right adjoint (respectively) to f^*.To develop (in the base category, say) the theory of the exact sequence of suspension, it is the existence of f_! that we need, and for this it suffices that \mathsf{Diag} contain the finite ordered sets (and even strictly less, if we insist).But I point out that the canonical sequences that we thus construct with the help of a single functor f_! and under the hypothesis that the derivator be “pointed” (i.e. the D(X) are pointed and the functors f^* are compatible with the zero objects) are only exact if we impose a suitable (left) “exactness axiom,” as part of the handful of axioms of a derivator;and dually for the exact sequence of cosuspension.These constructions still work not only in every category D(e), but also naturally in the D(X), for X in \mathsf{Diag}.(In fact, D(X) can be considered as the base category of an “induced derivator”

In {\mathcal{M}}, I also write D_{{\mathcal{M}}} instead of D_{({\mathcal{M}},W)}.It is a case that one might consider as “trivial,” but which however does not lack interest.Then D_{{\mathcal{M}}} is a derivator (satisfying all the axioms) provided only that {\mathcal{M}} is stable under small inductive and projective limits (one ensures the existences of the f_!, the other that of the f_*).In the case where {\mathcal{M}}=\mathsf{Set}, we get D(X)=X^\wedge, which is an important derivator in my eyes (no matter how trivial), and the “cohomological” properties of the arrows in \mathsf{Cat}, with respect to this derivator, are by no means something trivial.In the case where W is arbitrary, I also write D_W instead of D_{({\mathcal{M}},W)}, and it is rare that there is any fear of confusion.The main questions that I then pose is of course that of the existence of the functors f_! and f_*.Unlike you, I have no qualms here in supposing the category {\mathcal{M}} to be stable under all types of limits that we need, and thus (if we want to work with all of \mathsf{Cat}) stable under small inductive and projective limits.I would not be surprised if there were a theorem that says that every derivator on \mathsf{Cat} can be described by such a model category (up to equivalence of derivators), or at least as the filtrant inductive limit of such derivators.In this outline, I foresee an “algebra of derivators” (consisting of a certain number of fundamental operations within the 2-category of all derivators, on \mathsf{Cat}, say, as a domain), which would be the reflection of algebraic operations of a similar nature, which are carried out on the level of model categories.I have a feeling, from an allusion in your January letter, that you have some idea of intuition for this type of structure, and we can discuss it again.But I emphasise right away that, for me, the true purpose of operations on the level of model categories is to obtain operations on the associated derivators (or prederivators, to begin with).On this subject, a remark on the subject of the functoriality of the prederivator associated to a model category ({\mathcal{M}},W).It is clear

\mapsto \operatorname{D}(I,{\mathcal{A}})from the category \mathsf{Cat}, or from some sufficiently full category, such as that of finite categories or that of finite ordered sets, should suffice to give rise to all the essential structures of a “derived category” (still in limbo);even if, of course, one has to impose the necessary axioms (and that I ended clearing up last year) [9, Chapter I].We recover the original derived category, “naked,” by taking I=e (the point category).But it would be improper, in full rigour, to consider the more complete structure (that I now call a “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 topological algebra, 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 n-stacks on a topos (notions that still haven’t been defined to this day, except for n\leqslant 2).But for me, the “paradise lost” for topological algebra is by no means the eternal semi-simplicial category \Delta^\wedge, no matter how useful it might be, and even less so is it topological spaces (both of which live inside the 2-category of toposes, which is like a common envelope), but instead the category \mathsf{Cat} of small categories, thought of with a geometric eye by the set of intuitions,

Astonishingly rich, coming from toposes.Indeed, the toposes that have as sheaves of sets the {\mathcal{C}}^\wedge, with {\mathcal{C}} in \mathsf{Cat}, are by far the simplest of the known toposes, and it is because I believe this that I stressed so much the examples of these (“categorical”) toposes in SGA 4 IV [2].I now come to the definition, as I understand it, of a “prederivator” D — it being understood that the more delicate notion of “derivator” is deduced from this by imposing some quite natural axioms, the list of which I will give to you if you ask me.To develop an algebra of derivators (and first of all, prederivators), we must first fix a common “domain” for those that we will consider, i.e. a full subcategory \mathsf{Diag} of \mathsf{Cat}.The case which I prefer at the moment is that where \mathsf{Diag} is all of \mathsf{Cat}, in which case I interpret a derivator as being a sort of “theory of coefficients” (homological or cohomological or homotopical, they are all the same) on \mathsf{Cat}, where this category is thought of as a category 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 \mathsf{Diag} is the category of finite ordered sets, or even, at a push, an even more restricted category.But to be fully comfortable, one has to suppose sooner or later that the category \mathsf{Diag} (“category of indices” or “types of diagrams” for the derivators in question) is stable under the typical constructions on categories: finite products, subcategories, pushouts, even \underline{\operatorname{Hom}};and also, of course, taking the opposite category, which happens particularly often when passing from a statement to its dual, in particular.When we only talk about prederivators, in the the cases that I know of we can take the domain to be all of \mathsf{Cat}.It