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natural transformations in category theory
Central object of study in category theoryIn 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". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

Let be an module homomorphism of right modules. For every left module there is a natural map , form a natural transformation For every right module there is a natural map defined by , form a natural transformation .

Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. For any pointed topological space and positive integer there exists a group homomorphism

from the th homotopy group of to the th homology group of Both and are functors from the category Top* of pointed topological spaces to the category Grp of groups, and is a natural transformation from to .

Given an endofunctor, that is, a functor that maps a category to that same category, then there is a natural transformation from the identity functor to that functor, with components
For example, if is a field, then for every vector space over we have a "natural" injective linear map from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor.

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 a functor (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 and are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism , there is some with which it does not commute; in some cases a single automorphism works for all candidate isomorphisms while in other cases one must show how to construct a different 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 for example. Some authors distinguish notationally, using for a natural isomorphism and for an unnatural isomorphism, reserving for equality (usually equality of maps).

As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus. The homotopy groups of a product space are naturally the product of the homotopy groups of the components, with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement. However, the torus (which is abstractly a product of two circles) has fundamental group isomorphic to , but the splitting is not natural. Note the use of , , and :

This abstract isomorphism with a product is not natural, as some isomorphisms of do not preserve the product: the self homeomorphism of (thought of as the quotient space ) given by (geometrically a Dehn twist about one of the generating curves) acts as this matrix on (it's in the general linear group of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components". Naturality is a categorical notion, and requires being very precise about exactly what data is given – the torus as a space that happens to be a product (in the category of spaces and continuous maps) is different from the torus presented as a product (in the category of products of two spaces and continuous maps between the respective components).

Every finite dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces. There is in general no natural isomorphism between a finite dimensional vector space and its dual space. However, related categories (with additional structure and restrictions on the maps) do have a natural isomorphism, as described below. The dual space of a finite dimensional vector space is again a finite dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite dimensional vector spaces over a given field. However, in the absence of additional constraints (such as a requirement that maps preserve the chosen basis), the map from a space to its dual is not unique, and thus such an isomorphism requires a choice, and is "not natural". On the category of finite dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space (say, by choosing a basis for every vector space and taking the corresponding isomorphism), but this will not define a natural transformation. Intuitively this is because it required a choice, rigorously because any such choice of isomorphisms will not commute with, say, the zero map; see for detailed discussion. Starting from finite dimensional vector spaces (as objects) and the identity and dual functors, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from "all linear maps" to "linear maps that respect this structure". Explicitly, for each vector space, require that it comes with the data of an isomorphism to its dual, In other words, take as objects vector spaces with a nondegenerate bilinear form This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps that commute with the isomorphisms: or in other words, preserve the bilinear form: (These maps define the naturalizer of the isomorphisms.) The resulting category, with objects finite dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces. In this category (finite dimensional vector spaces with a nondegenerate bilinear form, maps linear transforms that respect the bilinear form), the dual of a map between vector spaces can be identified as a transpose. Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric (orthogonal matrices), symmetric and positive definite (inner product space), symmetric sesquilinear (Hermitian spaces), skew symmetric and totally isotropic (symplectic vector space), etc. – in all these categories a vector space is naturally identified with its dual, by the nondegenerate bilinear form.

Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.

If is an object of a locally small category , then the assignment defines a covariant functor This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of ). The natural transformations from a representable functor to an arbitrary functor are completely known and easy to describe; this is the content of the Yoneda lemma.

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations. The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.