PlanetPhysics/Categorical Diagrams Defined by Functors

Categorical Diagrams Defined by Functors

Any categorical diagram can be defined via a corresponding [[../TrivialGroupoid/|functor]] (associated with a [[../TrivialGroupoid/|diagram]] as shown by Mitchell, 1965, in ref. ). Such functors associated with diagrams are very useful in the categorical theory of [[../CategoricalGroupRepresentation/|representations]] as in the case of [[../CategoricalAlgebra/|categorical algebra]]. As a particuarly useful example in (commutative) homological algebra let us consider the case of an exact [[../HomologicalSequence2/|categorical sequence]] that has a correspondingly defined exact functor introduced for example in [[../AbelianCategory2/|abelian category]] theory.

Examples

Consider a scheme $\mathrm{\Sigma }$ as defined in ref. . Then one has the following short list of important examples of diagrams and functors:

1. Diagrams of adjoint situations: [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|adjoint functors]]

1. Equivalence of [[../Cod/|categories]]
2. [[../IsomorphismClass/|natural equivalence]] diagrams

1. Diagrams of [[../VariableCategory2/|natural transformations]]

1. Category of diagrams and 2-functors

1. [[../CategoricalGroupRepresentation/|monad]] on a category