Indexing categories of derivators

Question

It is not clear to me the role of the domain and target in the definition of prederivators.

For instance, the classical references put the domain as $\mathit{Dia}$, others as $\mathit{Cat}$ itself. Sometimes $\mathit{Dia}$ comes with some conditions, and sometimes not.

The target can also vary from the large CAT, or the 2-category of triangulated categories….

Grothendieck uses $$ D: \mathit{Dia}^\circ \rightarrow \mathit{Cat}. $$ On suppose tout au moins que $\mathit{Dia}$ contient les ensembles ordonnés finis, que $\mathit{Dia}$ est stable par limites finies, par sommes finies. Qu'avec toute catégorie $C$ et toute sous-catégorie strictement pleine ouverte (resp. fermée) elle contienne la sous-catégorie complémentaire.

Wikipedia says that the idea of this domain is to serve as "indices".

What prevents us from using not a 2-sub-category of $\mathit{Cat}$ but the complete category as domain and target?

Sorry if this question is too simple for the site. But can someone clean this up explaining the use of $\mathit{Dia}$ here?

Answer 1

The idea is to have more freedom of expression. No one wants to work with a fixed $Dia$ for ever. In particular, $Dia$ is made more general to allow us to change $Dia$ at will, according to our needs, including to make weird technical choices in some proof (for instance, if you want that (pre)derivators collectively form a reasonnable $2$-category without using Grothendieck universes, you could be happy to know that $Dia$ is small, or you might want to avoid (pre)derivators taking values in categories which are not locally small because you love the Yoneda Lemma and you do not know how to prove that your main example has this property for arbitrary shapes of diagrams yet). However, we can define $Cat$-indexed derivators and say that the restriction to some category of diagrams $Dia$ satisfy the axiom of derivators are satisfied, and this is fine most of the time. But there are properties which can proved using only certain shapes of diagrams and it is natural to restrict the theory to the fragment which only deals with them. A nice example is the universal property of bounded derived categories, mimicking Keller's approach via towers of triangulated categories. The proof only uses finite diagrams. It is formal to deduce an analogous universal property for the $Cat$-indexed derivator associated to an exact category $\mathcal{E}$ (associating to each $C$ the bounded derived categories of $\mathcal{E}$-valued functors defined on $C$). However, this is a typical example where I do not see how to deduce formally that the version for $Cat$-indexed derivators implies the version for derivators indexed by finite partially ordered sets (if you need that the assignment $X\mapsto\mathbb{D}(X\times C)$ is a (stable) derivator on finite partially ordered sets for any small category $C$, it is not easy to produce such a thing from a (stable) derivator derfined on finite partially ordered sets only). Therefore, in this case, the version for finite diagrams only seems to be a more powerful statement.

Answer 2

I think Mike's communicated the essential point. To belabor it, in most examples, if $D$ is a prederivator that anybody actually wants to talk about then $D(J)$ should be the homotopy category of $J$-shaped diagrams in some sort of homotopy theory. If this homotopy theory is something like, say, a fibration category, then you are never going to talk about the homotopy theory of diagrams indexed by some huge uncountable category, because you have no model for it. And you certainly can't prove that you have a derivator indexed by all small categories.

A derivator is a generalization of a complete and cocomplete category, so it's just a matter of parsimony to want it to have all Kan extensions that could be asked for, or every statement would start with "if $J$ is a category that's in the 2-category of diagrams that are all we're really talking about..." It's also possible to extend Dia to think about derivators indexed by higher categories, which are naturally produced for instance by complete and cocomplete $\infty$-categories, as suggested for instance by Raptis.

While most of the time, the choice of $\mathrm{Dia}$ is of limited consequence, not every argument can be immediately transferred away from $\mathrm{Cat}.$ For instance, the last I heard, nobody had quite managed to adapt Cisinski's theorem that the derivator (indexed by all small categories) of spaces is freely generated by a point to prove the analogous results for finite spaces and spectra.