categorical diagrams defined as 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. [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 $ \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

  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

Bibliography

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



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