In mathematics, specifically in category theory, an extranatural transformation is a generalization of the notion of natural transformation.
Let F : A × B o p × B → D and G : A × C o p × C → D two functors of categories. A family η ( a , b , c ) : F ( a , b , b ) → G ( a , c , c ) is said to be natural in a and extranatural in b and c if the following holds:
η ( − , b , c ) is a natural transformation (in the usual sense).(extranaturality in b) ∀ ( g : b → b ′ ) ∈ M o r B , ∀ a ∈ A , ∀ c ∈ C the following diagram commutes(extranaturality in c) ∀ ( h : c → c ′ ) ∈ M o r C , ∀ a ∈ A , ∀ b ∈ B the following diagram commutesExtranatural transformations can be used to define wedges and thereby ends (dually co-wedges and co-ends), by setting F (dually G ) constant.
Extranatural transformations can be defined in terms of Dinatural transformations.
