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.

Examples

Consider a scheme ${\displaystyle \Sigma }$ 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

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

All Sources

^[1]

References

1. 1 2 3 Barry Mitchell., Theory of Categories. , Academic Press: New York and London (1965), pp.65-70.