Editing Natural transformation (section)

===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).