In mathematics, derivators are a proposed framework[1][2]pg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations 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Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks.
They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages.
Essentially the same concept was introduced (apparently independently) by Alex Heller.
[3] 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.
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.
which is a category with two objects and one non-identity arrow, and a functor
(and satisfying the right hypotheses), we should have an associated functor
where the target object is unique up to weak equivalence in
Derivators are able to encode this kind of information and provide a diagram calculus to use in derived categories and homotopy theory.
Formally, a prederivator
{\displaystyle \mathbb {D} :{\text{Ind}}^{op}\to {\text{CAT}}}
Typically such 2-functors come from considering the categories
is called the category of coefficients.
could be the category of small categories which are filtered, whose objects can be thought of as the indexing sets for a filtered colimit.
This is called the inverse image functor.
In the motivating example, this is just precompositition, so given a functor
is a suitable class of weak equivalences in a category
There are a number of examples of indexing categories which can be used in this construction Derivators are then the axiomatization of prederivators which come equipped with adjoint functors where
should correspond to inverse limits,