PlanetPhysics/Categorical Diagrams Defined by Functors
Categorical Diagrams Defined by Functors
editAny 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
editConsider 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