Natural Transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e. the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most basic notions of category theory and consequently appear in the majority of its applications.

Read more about Natural TransformationDefinition, Unnatural Isomorphism, Operations With Natural Transformations, Functor Categories, Yoneda Lemma, Historical Notes, Symbols Used

Other articles related to "natural transformation, natural transformations, natural":

Comma Category - Examples of Use - Natural Transformations
... which defines morphisms in with is identical to the diagram which defines a natural transformation ... The difference between the two notions is that a natural transformation is a particular collection of morphisms of type of the form, while objects of the comma category ... This is described succinctly by an observation by Huq that a natural transformation, with, corresponds to a functor which maps each object to and maps each morphism to ...
Monoidal Natural Transformation
... A monoidal natural transformation between those functors is a natural transformation between the underlying functors such that the diagrams and commute for ... A symmetric monoidal natural transformation is a monoidal natural transformation between symmetric monoidal functors ...
Adjoint Functors - Adjunctions in Full - Universal Morphisms Induce Hom-set Adjunction
... Construct a functor and a natural transformation ... factorization implies the commuting diagram of natural transformations, so is a natural transformation ... Construct a natural isomorphism ...
Representable Functor - Universal Elements
... According to Yoneda's lemma, natural transformations from Hom(A,–) to F are in one-to-one correspondence with the elements of F(A) ... Given a natural transformation Φ Hom(A,–) → F the corresponding element u ∈ F(A) is given by Conversely, given any element u ∈ F(A) we may define a ... In order to get a representation of F we want to know when the natural transformation induced by u is an isomorphism ...

Famous quotes containing the word natural:

    The natural historian is not a fisherman who prays for cloudy days and good luck merely; but as fishing has been styled “a contemplative man’s recreation,” introducing him profitably to woods and water, so the fruit of the naturalist’s observations is not in new genera or species, but in new contemplations still, and science is only a more contemplative man’s recreation.
    Henry David Thoreau (1817–1862)