In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories 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 Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages.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.
Motivation
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 diagramwhich is a category with two objects and one non-identity arrow and a functorto a category with a class of weak-equivalences, and satisfying the right hypotheses, should have an associated functorwhere the target object is unique upto 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.
Definition
Prederivators
Formally, a prederivator is a 2-functorfrom a suitable 2-category of indices to the category of categories. Typically such 2-functors come from considering the categories where is called the category of coefficients. For example, 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 diagramsdenote byThis is called the inverse image functor. In the motivating example, this is just precompositition, so given a functor there is an associated functor. Note these 2-functors could be taken to bewhere is a suitable class of weak equivalences in a category.
Indexing categories
There are a number of examples of indexing categories used in this construction
The 2-category of finite categories, so the objects are categories whose collection of objects are finite sets.
The ordinal category can be categorified into a two category, where the objects are categories with one object, and the functors come form the arrows in the ordinal category.
Another option is to just use the category of small categories.
In addition, associated to any topological space is a category which could be used as the indexing category.
This can be generalized to any topos, so the indexing category is the underlying site.
Derivators
Derivators are then the axiomatization of prederivators which come equipped with adjoint functorswhere is left adjoint to and so on. Heuristically, should correspond to inverse limits, to colimits.
