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So I was reading through a bit of literature on derivators, and I noticed that two of the axioms (Der 3 and Der 4) assert that our derivators should have all homotopy kan extensions along functors belonging to $Dia$, and that these are effectively pointwise.
Is there any way to weaken this notion by removing this very strong bicompleteness condition on derivators without completely wrecking the theory? In particular, something nice would be the ability to have all quasicategories have associated derivators, if we could do it.
Well, I think a prederivator that doesn’t at least have finite limits or colimits is not going to be nearly as useful a structure as a general quasicategory; you’ve thrown away enough structure that it’s hard to do very much. But yes, it might be nice to have a word for a prederivator that satisfies (Der1), (Der2), and (Der5) only. I would myself probably have been inclined to say “derivator” for that and “bicomplete derivator” when (Der3) and (Der4) are added, but Grothendieck’s preferences were otherwise. Any suggestions?
Maybe either semiderivator or quasiderivator? (Cisinski calls derivators missing either holims or hocolims “right-weak” and “left-weak” respectively)
By the way, I think that for the theory to be effectively equivalent to quasicategories, it seems like we would want the smallest class of prederivators that contains all of the derivators representable by homotopical categories à la Dwyer-Hirschhorn-Kan-Smith, since these (I think I saw this in Joyal’s notes) effectively represent every quasicategory (I think he calls this something like a “presentation” of a quasicategory).
By the way, how does the theory fail in the absence of these assumptions? Is it possible that in the classical theory, we can embed every prederivator into a bicomplete derivator, then deduce the holim and hocolims that live inside of the original prederivator by a notion of homotopy representability?
By the way, I remember that you were griping a bit about how all of the literature was in French (as was I), but I found this paper in English (written by a Ph. D. student at Bonn) that is quite readable compared to the French material (yes, the French stuff is alright as long as you know roughly what’s going on in the first place!)
What might be useful is to see what the local properties of a derivator are, then require them for those holims/hocolims that do exist.
I could live with semiderivator or quasiderivator. I don’t think you’re ever going to get a theory that’s “effectively equivalent” to quasicategories, though; there’s just too much information being lost. I’m not sure of an original reference for the fact that all (∞,1)-categories can be represented as localizations of categories with weak equivalences; most recently and comprehensively there is the paper of Barwick and Kan on what they call “relative categories.”
Regarding how the theory fails, I don’t actually know what you could do. All the theory of derivators that I know of uses limits and colimits; what would you want to do with them? I’m sure you could get away with assuming only the existence of the particular limits and colimits you need for any particular application, though. I think it should be easy to define what you mean by a particular limit or colimit existing in a prederivator; just say that the relevant restriction functor has an adjoint at that particular object, which is respected by all homotopy exact squares. Is that what you mean by “local” properties?
@Harry The obvious retort is try to learn more French.
As a PG I used my rudimentary school French plus a dictionary to write a rough translation of parts of SGA4. I learnt quite a lot of vocabulary that way plus some topos theory! I was slow to start with, but I got better slowly. Mathematical French is relatively easy as is mathematical German or Flemish or any of the main western European languages. I cannot manage Russian.. and was never trying for more than a working translation for my purposes. i.e. I use the translated proof to build my own attempt, (with SGA4 the proofs are sketchy sometimes and are incorrect in detail moderately often as I remember them.)
just say that the relevant restriction functor has an adjoint at that particular object, which is respected by all homotopy exact squares.
I’ve spelled out what I mean by this at local Beck-Chevalley condition, and added a definition of semiderivator.
Coolio!
I wonder if there’s anything nice that’s true about these semiderivators. I bet Cisinski would know. Maybe I’ll e-mail him.
One thing that Ross Street has pointed out in a couple of places is that axioms (Der3) and (Der4) can equivalently be phrased as saying that D admits pointwise Kan extensions along morphisms between representables in the presheaf 2-category $[Dia^{op},CAT]$. (One has to define “pointwise” in a suitable way to make sense internally in a 2-category; the result turns out to be equivalent to (Der4).)
Maltsiniotis was saying that the derivators are rather special and that a homotopy theory/derived functors story in his opinion should be thought in three stages from a more general to more special
1) Cisinski’s catégories dérivables, cf.
2) derivators
3) model categories
We had not discussed the more general stage 1 in $n$Lab so far. I know nothing about them.
It looks to me like Cisinski’s “derivable categories” are another one more of the weaker-than-model-categories structures that one can put on a category with weak equivalences that still enables one to do homotopy theory. We have certainly talked about other such structures, like categories of fibrant objects.
Surely, this is true. There is a construction of a derivator from a derivable category, I am not sure what it looses. What I like is that it is non-symmetric. It is weaker than categories of fibrant objects as the “left” derivable category $C$ contain Brown’s categories as category $C_f$ of fibrant objects in $C$.
What I like is that it is non-symmetric.
This is somewhat analogous to the situation with Rosenberg-right/left-exact categories which he uses in his nonabelian homological algebra. Rosenberg requires a structure of subcanonical singleton (co)topology, where the distinguished epis or monos (depending on right or left) are one sided (non-symmetric) version of structure of Quillen exact category with admissible epis and admissible monos. Rosenberg’s work subsequently defines (1-categorical, nonadditive) versions of stable categories: nonadditive presuspended and nonadditive triangulated categories. Notice that he does not require $0$-object, just initial or jkust final, depending on side, not zero; but he has chain complexes, notion of projective resolutions and alike in that setup. It is tempting to compare his half exact categories to Cisinski’s formalism.
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