Exactors in Category Theory: A Guide to Understanding Exactness in Functors

Exactors are a way to define a notion of "exactness" for a functor, which can be used to study the properties of the functor.

An exactor is a pair of a functor and a natural transformation between it and the identity functor. The idea is that the functor is "exact" in the sense that it preserves some kind of structure, such as a group or ring structure, and the natural transformation is a way to measure how well the functor preserves this structure.

For example, if we have a functor F: Grp -> Ab, where Grp is the category of groups and Ab is the category of abelian groups, then an exactor for F might be a pair (F, ε), where ε is a natural transformation from F to the identity functor Id_Ab, such that ε(g) is a homomorphism from F(g) to g for all objects g in Grp. This means that F preserves the group structure of the objects in Grp, and ε measures how well F preserves this structure.

Exactors have many applications in category theory, including the study of limits and colimits, the definition of derived functors, and the study of natural transformations between functors. They are also closely related to other important concepts in category theory, such as exact sequences and triangles.