Definition. Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and let $S,T:\mathcal{C}\to\mathcal{D}$ be covariant functors. Then suppose that for every object $A$ in $\mathcal{C}$ one has a morphism $\eta_A : S(A) \to T(A) $ in $\mathcal{D}$ such that for every morphism $\alpha: A \to B$ in $\mathcal{C}$ the following

is commutative. Then we variously write $$\eta: S \dot{\to} T \quad\mbox{ or }\quad \eta: S\Rightarrow T\quad \mbox{ or } \quad \eta:S\to T$$ and call $\eta$ a natural trasformation from $S$ to $T$ .

One may think of a natural transformation $\eta:S\to T$ as a `function' from the class of objects of $\mathcal{C}$ to the class of morphisms of $\mathcal{D}$ .

As a first example, for every functor $S:\mathcal{C}\to \mathcal{D}$ , we can associate the natural transformation $1_S: S\to S$ (the identity natural transformation on $S$ ) that assigns every object $A$ of $\mathcal{C}$ , the corresponding identity morphism $1_{S(A)}$ .

Natural transformations are composed in a similar manner to morphisms, but they are nevertheless defined as correspondences between both objects and morphisms as shown in the square commutative diagram depicted above.

More precisely, given three functors $R,S,T:\mathcal{C}\to \mathcal{D}$ , and two natural transformations, $\tau:R\to S$ and $\eta:S\to T$ , we define the composition of $\tau$ with $\eta$ , written $\eta \bullet \tau$ , as a class of morphisms in $\mathcal{D}$ given by $$(\eta\bullet \tau)_A := \eta_A\circ \tau_A,$$ for every object $A$ in $\mathcal{C}$ . It is easy to see that $\eta\bullet \tau$ is a natural transformation, since we may ``compose'' two commutative squares and obtain a third one:

Remarks.

• Natural transformations arise frequently in mathematics. Here's a ``concrete'' example of a natural transformation: the boundary map $H_n(X,A)\to H_{n-1}(A)$ in a homology theory. By definition, every morphism of presheaves is a natural transformation. More prosaically, the determinant $\det:\GL_n\dot{\to}(\ )^*$ is natural.
• Natural transformations are sometimes called also functorial morphisms especially in applications related to the category theory development line pursued by Charles Ehresmann and the `Nicolas Bourbaki' group; this is also a natural translation of the same concept from French, viz. (ref. [3]).

Bibliography

1

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

2

S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer-Verlag, 1997.

3

C. Ehresmann, Catégories et Structures. Dunod: Paris , 1965.

4

C. Ehresmann, Catégories doubles des quintettes: applications covariantes , C.R.A.S. Paris, 256: 1891-1894, 1963.

5

S. Eilenberg and S. Mac Lane, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294, 1945.

6

B. Mitchell., Theory of Categories, Academic Press: New York and London.