# PlanetPhysics/Categorical Diagrams Defined by Functors

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### Categorical Diagrams Defined by Functors

[edit | edit source]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

[edit | edit source]Consider a scheme as defined in ref. ^{[1]}. Then one has the following short list of important examples of diagrams and functors:

- Diagrams of adjoint situations: adjoint functors

- Equivalence of categories
- natural equivalence diagrams

- Diagrams of natural transformations

- Category of diagrams and 2-functors

- monad on a category

## All Sources

[edit | edit source]^{[1]}

## References

[edit | edit source]- ↑
^{1.0}^{1.1}^{1.2}Barry Mitchell.,*Theory of Categories.*, Academic Press: New York and London (1965), pp.65-70.