Update grothendieck-thomason-91-04-02.Rmd

rmd/grothendieck-thomason-91-04-02.Rmd

@@ -136,9 +136,9 @@ The same idea had the appearance of working for the non-commutative variants of
 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.
+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 $n\leq2$).
-But for me, the "paradise lost" for algebraic topology 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 $\Cat$ of small categories, thought of with a geometric eye by the set of intuitions, astonishingly rich, coming from toposes.
+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 $\Cat$ 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 $\cal{C}^\wedge$, with $\cal{C}$ in $\Cat$, 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.
@@ -291,7 +291,7 @@ I do not know if it is reasonable to expect, for the $2$-functor above ($***$),
 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...)^[*[Editor] This problem was studied by O. Renaudin [@11].*]
 
 But I have to return to the case where we are given a fixed model category $(\cal{M},W)$, and to the big question of the existence of the functors $f_!$ and $f_*$.
-Technically speaking, it is here, evidently, that we find one of the most crucial questions for the development of algebraic topology, as I consider it.
+Technically speaking, it is here, evidently, that we find one of the most crucial questions for the development of topological algebra, as I consider it.
 But for this fundamental question, I have only fragments of a good answer, and they are manifestly unsatisfying (and probably also insufficient in the long run).
 Taking the case where $\Diag$ is equal to $\Cat$: I admit (to my shame!) that I have not even constructed an example of a model category that is stable under small (inductive and projective) limits such that the functors $f_!$ and $f_*$ do not exist for every arrow $f$ in $\Cat$, for the associated prederivator.
 In fact, I don't even expect them to exist, even if we make hypotheses of the type "$W$ is stable under filtrant inductive limits, and the category $\cal{M}$ is accessible and $W$ is an accessible subset of $\Mor(\cal{M})$" (hypotheses which seem to me to be relatively trivial).