Prederivators

Formally, a prederivator ${\displaystyle \mathbb {D} }$ is a 2-functor

${\displaystyle \mathbb {D}$ :{\text{Ind}}^{op}\to {\text{CAT}}}

from a suitable 2-category of indices to the category of categories. Typically such 2-functors come from considering the categories ${\displaystyle {\underline {\text{Hom}}}(I^{op},A)}$ where ${\displaystyle A}$ is called the category of coefficients. For example, ${\displaystyle {\text{Ind}}}$ 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

${\displaystyle f:I\to J}$

denote ${\displaystyle f^{*}}$ by

${\displaystyle f^{*}:\mathbb {D} (J)\to \mathbb {D} (I)}$

This is called the inverse image functor. In the motivating example, this is just precompositition, so given a functor ${\displaystyle F_{I}\in {\underline {\text{Hom}}}(I^{op},A)}$ there is an associated functor ${\displaystyle F_{J}=F_{I}\circ f}$. Note these 2-functors could be taken to be

${\displaystyle {\underline {\text{Hom}}}(-,A[W^{-1}])}$

where ${\displaystyle W}$ is a suitable class of weak equivalences in a category ${\displaystyle A}$.

Derivators

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

${\displaystyle f^{?}\dashv f_{!}\dashv f^{*}\dashv f_{*}\dashv f^{!}}$

where ${\displaystyle f_{!}}$ is left adjoint to ${\displaystyle f^{*}}$ and so on. Heuristically, ${\displaystyle f_{*}}$ should correspond to inverse limits, ${\displaystyle f_{!}}$ to colimits.