Derivator

In mathematics, derivators are a proposed new frameworkpg 190-195 for homological algebra giving a framework for non-abelian homological algebra and various generalisations of it. They were introduced to address the deficiencies of derived categories (such as the non-functoriality of the cone construction) and provide at the same time a language for homotopical algebra.
Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript "Pursuing Stacks". They were then further developed by him in the huge unpublished 1991 manuscript "Les Dérivateurs" of almost 2000 pages.
The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller, Franke, Keller and Groth.
Motivations.
One of the motivating reasons for considering derivators is the lack of functoriality with the cone construction with triangulated categories. Derivators are able to solve this problem, and solve the inclusion of general homotopy colimits, by keeping track of all possible diagrams in a category with weak equivalences and their relations between each other. Heuristcally, given the diagramformula_1which is a category with two objects and one non-identity arrow and a functorformula_2to a category formula_3 with a class of weak-equivalences formula_4, and satisfying the right hypotheses, should have an associated functorformula_5where the target object is unique up to weak equivalence in formula_6. Derivators are able to encode this kind of information and provide a diagram calculus to use in derived categories and homotopy theory.
Definition.
Prederivators.
Formally, a prederivator formula_7 is a 2-functorformula_8from a suitable 2-category of indices to the category of categories. Typically such 2-functors come from considering the categories formula_9 where formula_3 is called the category of coefficients. For example, formula_11 could be the category of small categories which are filtered, whose objects can be thought of as the indexing sets for a filtered colimit. Then, given a morphism of diagramsformula_12denote formula_13 byformula_14This is called the inverse image functor. In the motivating example, this is just precompositition, so given a functor formula_15 there is an associated functor formula_16. Note these 2-functors could be taken to beformula_17where formula_4 is a suitable class of weak equivalences in a category formula_3.
Indexing categories.
There are a number of examples of indexing categories which can be used in this construction
Derivators.
Derivators are then the axiomatization of prederivators which come equipped with adjoint functorsformula_27where formula_28 is left adjoint to formula_13 and so on. Heuristically, formula_30 should correspond to inverse limits, formula_28 to colimits.