0.1 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. [1]). 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.

0.2 Examples

Consider a scheme Σ as defined in ref. [1]. Then one has the following short list of important examples of diagrams and functors:

1. Diagrams of adjoint situations: Adjoint functors
2. Equivalence of categories
3. Natural equivalence diagrams
4. Diagrams of natural transformations
5. Category of diagrams and 2-functors
6. monad on a category

References