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I would like to enter the world of derivators. We can find a little history here and there about the limitations of triangulated categories and the motivation to enhance them, but also to compute homotopy limits and colimits and others. There is also a relation with the theory of higher categories. History also says that Grothendieck, Heller, and in some sense Franke found the notion respectively for their own needs.

The references are varied regarding what axioms should one to impose on prederivators

Can you give a high explanation (motivation) for the use of the axioms of derivators?

The background: category theory and some (not so serious) categorical homotopy theory.

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    $\begingroup$ The nLab article you link to has some explanatory remarks about the axioms. Do you want us to explain those remarks, or...? $\endgroup$
    – Zhen Lin
    Commented Oct 16, 2022 at 22:18
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    $\begingroup$ All of the axioms are designed to hone in on those prederivators which look something like the 2-functors of homotopy categories of diagrams in some abstract homotopy theory, that is, $J\mapsto \mathrm{Ho}(C^J),$ where $C$ is a nice model category or $\infty$-category. If that much is already clear then please be more specific about what you’re looking for. $\endgroup$
    – Kevin Carlson
    Commented Oct 18, 2022 at 20:09
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    $\begingroup$ @The reason nLab was not enough was that the notion is well motivated, but I would like to know about the particular choice of axioms. You can motivate a notion, for example, a group (resp. topological space) as a theory of symmetry (with localization) but the choice of axioms seems to be part of an experiment (Hausdorff axioms, open subsets,...). I am interested in a motivation for the choice of axioms. $\endgroup$
    – user234212323
    Commented Oct 19, 2022 at 18:28
  • $\begingroup$ I'm wondering if there was anything in Pursuing Stacks that prefigured Grothendieck's Les Derivateurs. I have a vague memory there might be, but it was a long time ago I read PS through. Grothendieck would certainly discuss his thought process, because PS was more like a research diary than a finished article, which LD is more in the style of. $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2022 at 8:08
  • $\begingroup$ Aha, I remembered correctly. Section 69 of Pursuing Stacks discusses derivators, and the example given is that triangulated categories were not enough, and one should consider, associated to an abelian category $\mathcal{A}$, for each 'diagram shape' $I$, the derived category $D(Hom(I,\mathcal{A}))$ of the abelian category of $I$-shaped diagrams in $\mathcal{A}$, and this data was to encode the missing limits and colimits in the ordinary derived category $D(\mathcal{A})$. This was the inspiration to consider the analogue for the homotopy category of an arbitrary localiser... $\endgroup$
    – David Roberts ♦
    Commented Oct 24, 2022 at 9:27

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With the comments having clarified the question a bit, let me just say that the a priori motivation for these particular axioms was just whatever was going through Grothendieck’s and Heller’s heads when they wanted to build something that looks like a 2-functor of categories of diagrams and Kan extensions between them. Certainly (Der3),(Der4) (the mere existence and pointwise nature of the Kan extensions) are unavoidable for this, while (Der1) (preservation of coproducts) might fall out of considering the very little extent to which a derivator might be like a stack. (Der2) (conservatively of the underlying diagram functors) is less obviously unavoidable but certainly makes you feel like you’re looking at diagrams, and is one way of guaranteeing that every diagram in a derivator can be built out of constant diagrams, so that a derivator is really like a refinement of its base.

(Der5) (weak quotientness of the underlying diagram functor for diagrams with free domain) is probably the least obvious, but is well motivated by the usefulness of distinguished weak colimits in triangulated category theory, and some parts of unstable homotopy theory. It is also not necessary for the strongest a priori justification of the other axioms: Cisinski’s theorem that the derivator of spaces is freely generated by the point. In particular, note that this theorem fails without (Der1) and (Der2), and is senseless without 3 and 4.

(Der5) is necessary to consider which derivators are representable by $\infty$-categories or model categories, since such derivators certainly satisfy (Der5). One goal of my thesis was to study the extent to which it’s sufficient. The answer is "not quite", but a similar axiom saying that a derivator respects all lax pushouts, rather than merely the lax pushout $0\to 1,$ is a sufficient replacement. The only other missing ingredients are preservation of certain coinverters, or localizations, a type of 2-colimit whose preservation fits in nicely with (Der1) but was not originally noticed, as well as various technical details about size.

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