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Wikipedia / Derivator

In mathematics, derivators are a proposed frameworkpg 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Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript Pursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscript Les Dérivateurs of almost 2000 pages.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.

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. Heuristically, given the diagramwhich 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. 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 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. Then, given a morphism of diagrams byThis is called the inverse image. Note these 2-functors could be taken to be.

Indexing categories

There are a number of examples of indexing categories which can be used in this construction

  • of finite categories, so the objects are categories whose collection of objects are finite sets.

  • can be categorified into a two category, where the objects are categories with one object, and the functors come from the arrows in the ordinal category.

  • Another option is to just use the category of small categories.

  • which could be used as the indexing category.

  • Moreover, the sites underlying the Zariski, Etale, etc., Grothendieck topoi for some scheme or algebraic space along with their morphisms can be used for the indexing category

  • , so the indexing category is the underlying site.

Derivators

Derivators are then the axiomatization of prederivators which come equipped with adjoint functors

to colimits.

References

Bibliography