Page "Natural transformation" Paragraph 25

The notion of a natural transformation is categorical, and states ( informally ) that a particular map between functors can be done consistently over an entire category.

Informally, a particular map ( esp.

an isomorphism ) between individual objects ( not entire categories ) is referred to as a " natural isomorphism ", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors ; formalizing this intuition was a motivating factor in the development of category theory.

Conversely, a particular map between particular objects may be called an unnatural isomorphism ( or " this isomorphism is not natural ") if the map cannot be extended to a natural transformation on the entire category.

Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).

More strongly, if one wishes to prove that X and G ( X ) are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism η, there is some A with which it does not commute ; in some cases a single automorphism A works for all candidate isomorphisms η, while in other cases one must show how to construct a different A < sub > η </ sub > for each isomorphism.

The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance.