Editing Natural transformation (section)

== Unnatural isomorphism ==
{{see also|Canonical map}}
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 "an isomorphism that is not natural") if the map cannot be extended to a natural transformation on the entire category. Given an object <math>X,</math> a functor <math>G</math> (taking for simplicity the first functor to be the identity) and an isomorphism <math>\eta\colon X \to G(X),</math> proof of unnaturality is most easily shown by giving an automorphism <math>A\colon X \to X</math> that does not commute with this isomorphism (so <math>\eta \circ A \neq G(A) \circ \eta</math>). More strongly, if one wishes to prove that <math>X</math> and <math>G(X)</math> are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for ''any'' isomorphism <math>\eta</math>, there is some <math>A</math> with which it does not commute; in some cases a single automorphism <math>A</math> works for all candidate isomorphisms <math>\eta</math> while in other cases one must show how to construct a different <math>A_\eta</math> 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 {{section link|Structure theorem for finitely generated modules over a principal ideal domain|Uniqueness}} for example.

Some authors distinguish notationally, using <math>\cong</math> for a natural isomorphism and <math>\approx</math> for an unnatural isomorphism, reserving <math>= </math> for equality (usually equality of maps).

===Example: fundamental group of torus===
As an example of the distinction between the functorial statement and individual objects, consider [[homotopy group]]s of a product space, specifically the fundamental group of the torus.

The [[homotopy group]]s of a product space are naturally the product of the homotopy groups of the components, <math>\pi_n((X,x_0) \times (Y,y_0)) \cong \pi_n((X,x_0)) \times \pi_n((Y,y_0)),</math> 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 <math>Z^2</math>, but the splitting <math>\pi_1(T,t_0) \approx \mathbf{Z} \times \mathbf{Z}</math> is not natural. Note the use of <math>\approx</math>, <math>\cong</math>, and <math>=</math>:{{efn|1='''Z'''<sup>''n''</sup> could be defined as the ''n''-fold product of '''Z''', or as the product of '''Z'''<sup>''n''&nbsp;&minus;&nbsp;1</sup> and '''Z''', which are subtly different sets (though they can be naturally identified, which would be notated as ≅). Here we've fixed a definition, and in any case they coincide for ''n''&nbsp;=&nbsp;2.}}
:<math>\pi_1(T,t_0) \approx \pi_1(S^1,x_0) \times \pi_1(S^1,y_0) \cong \mathbf{Z} \times \mathbf{Z} = \mathbf{Z}^2.</math>
This abstract isomorphism with a product is not natural, as some isomorphisms of <math>T</math> do not preserve the product: the self-homeomorphism of <math>T</math> (thought of as the [[Quotient space (topology)|quotient space]] <math>R^2/\mathbb{Z}^2</math>) given by <math>\left(\begin{smallmatrix}1 & 1\\0 & 1\end{smallmatrix}\right)</math> (geometrically a [[Dehn twist]] about one of the generating curves) acts as this matrix on <math>\mathbb{Z}^2</math> (it's in the [[general linear group]] <math>\text{GL}(\mathbb{Z}, 2)</math> 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 <math>(T,t_0) = (S^1,x_0) \times (S^1,y_0)</math> – 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).

===Example: dual of a finite-dimensional vector space===
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.<ref>{{harv|Mac Lane|Birkhoff|1999|loc=§VI.4}}</ref> 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 {{harv|Mac Lane|Birkhoff|1999|loc=§VI.4}} 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, <math>\eta_V\colon V \to V^*</math>. In other words, take as objects vector spaces with a [[nondegenerate bilinear form]] <math>b_V\colon V \times V \to K</math>. This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps <math>T\colon V \to U</math> that commute with the isomorphisms: <math>T^*(\eta_{U}(T(v))) = \eta_{V}(v)</math> or in other words, preserve the bilinear form: <math>b_{U}(T(v),T(w))=b_V(v,w)</math>. (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 space]]s), 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.