Natural Transformation - Unnatural Isomorphism

Unnatural Isomorphism

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_η 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.

This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain#Uniqueness for example.

Some authors distinguish notationally, using ≅ for a natural isomorphism and ≈ for an unnatural isomorphism, reserving = for equality (usually equality of maps).