PlanetPhysics/Categorical Diagrams Defined by Functors

Categorical Diagrams Defined by Functors
Any categorical diagram can be defined via  a corresponding functor (associated with a diagram as shown by Mitchell, 1965, in ref. ). Such functors associated with diagrams are very useful in the categorical theory of representations as in the case of categorical algebra. As a particuarly useful example in (commutative) homological algebra let us consider the case of an exact categorical sequence that has a correspondingly defined exact functor introduced for example in abelian category theory.

Examples
Consider a scheme $$\Sigma$$ as defined in ref. . Then one has the following short list of important examples of diagrams and functors:


 * 1) Diagrams of adjoint situations: adjoint functors


 * 1) Equivalence of categories
 * 2) natural equivalence diagrams


 * 1) Diagrams of natural transformations


 * 1) Category of diagrams and 2-functors


 * 1) monad on a category