Talk:Natural transformation

Would you give an example of a non-natural isomorphism to the article? -- Taku 00:26, 10 November 2005 (UTC)

It looks like it has been added... Marc Harper 15:08, 26 March 2006 (UTC)

An isomorphism between a vector space and its dual V* is perhaps the commonest example of a non-natural isomorphism. Is there a sense in which it is a theorem that there is no natural isomorphism between V and V* ?Daqu 04:46, 21 February 2006 (UTC)

I removed the bit saying it was a 'counterexample'.

metterklume

It's probably not too hard to show that there can be no transformations between functors of different variance. -lethe ^{talk +} 05:25, 21 February 2006 (UTC)

example for fun

I wrote this example natural transformation for the benefit of a friend in my user space. He suggested that it was well-enough presented to warrant inclusion in this article, or else perhaps in its own article. I don't really think so; it seems to me to be too tutorial-ish to be encyclopedic. I post it on the talk page, and you can decide whether it has a place in this article (or any other).

We want to verify the equation

$\tau_A\circ \mathcal{P}(f) = f^*\circ\tau_B$

where τ_C: P(C) → 2^C is the map which sends any subset of the set C to the characteristic function on that subset, i.e.

$\tau_C(U)=\chi_U,$

where χ_U is given by

$\chi_U(c) = \begin{cases} 1 & c \in U\\ 0 & c \not\in U \end{cases}$

for any subset U ⊆ C and any element c ∈ C. To verify the equation, let both sides act on some subset S ⊆ B. We have

$\mathcal{P}(f)(S) = f^{-1}(S)$

by the definition of the powerset functor, and so

$\tau_A(\mathcal{P}(f)(S)) =\chi_{f^{-1}(S)}.$

On the right-hand side of the equation, we have

$\tau_B(S) = \chi_S$

and recall that f* is the pullback by f induced by the contravariant hom-functor; it acts on maps by multiplication on the right:

$f^*(\chi_S)=\chi_S\circ f.$

So it remains to check the equality

$\chi_{f^{-1}(S)} = \chi_S\circ f.$

To verify this equation, act both maps in 2^A on an arbitrary element a ∈ A.

$\chi_{f^{-1}(S)}(a) = \begin{cases} 1 & a\in f^{-1}(S)\\ 0 & a \not\in f^{-1}(S) \end{cases}$

$(\chi_S\circ f)(a)= \begin{cases} 1 & f(a)\in S\\ 0 & f(a) \not\in S \end{cases}$

Since a ∈ f^–1(S) iff f(a) ∈ S, these maps are equal.

It's better than the "every group is naturally isomorphic to its opposite group"... —Preceding unsigned comment added by 130.126.108.212 (talk) 19:12, 9 November 2007 (UTC)

natural isomorphism

I removed this paragraph from the article. I think it's wrong, but I don't know enough math to be sure of it. Paisa (talk) 02:23, 21 December 2007 (UTC)

A natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1_G and εη = 1_F (where 1_F : F → F is the natural transformation assigning to every object X the identity morphism on F(X)).

I don't have CWM handy, but Mac Lane and Moerdijk in Sheaves and Geometry in Logic on page 13 call a natural transformation η : F → G a natural isomorphism if it is componentwise an isomophism, that is, if η_X is an isomorphism for each object X. (This is the definition currently in the article.) The Mac Lane–Moerdijk definition is equivalent to the one above. If there exists a natural transformation ε as above, then the compositional identities imply that η_X and ε_X are inverse morphisms. Thus η is componentwise an isomorphism, hence a natural isomorphism. Conversely, given a natural isomorphism η, define ε by the rule ε_X := η_X^–1 for each object X. One can verify that this definition yields a natural transformation satisfying the compositional identities above. I did not find an explicit comment in Mac Lane–Moerdijk acknowledging this equivalent definition. Michael Slone (talk) 04:09, 21 December 2007 (UTC)

Actually, I think that the definition

A natural transformation η : F → G is a natural isomorphism if and only if there exists a natural transformation ε : G → F such that ηε = 1_G and εη = 1_F (where 1_F : F → F is the natural transformation assigning to every object X the identity morphism on F(X)).

is better than the one that requires each η_X to be an isomorphism, as it gives the right generalisations to terms such as natural retraction, for which the two mechanisms of definition give non-equivalent concepts. I.e. a natural retraction should be a natural transformation with a right-inverse natural transformation, not a natural transformation η for which each η_X is a retraction (and which retractions may not necessarily fit together to make a right inverse natural transformation, unlike the case with isomorphisms). If nobody objects I may therefore update the article with this definition, pointing out the equivalence with the other definition for isomorphisms, and indicating the generalisations to retraction, coretraction, etc. Rfs2 (talk) 09:38, 7 October 2011 (UTC)

Natural map vs transformation

I think there's an informal notion of natural map that's not a natural transformation. I posted a section in Talk:Eilenberg-Steenrod axioms. Can someone explain how exactly the connecting homomorphism is a transformation? Money is tight (talk) 05:50, 12 January 2011 (UTC)

opposite group of abelian group

For all abelian groups X, Y and Z we have a group isomorphism

Hom(X ⊗ Y, Z) → Hom(X, Hom(Y, Z)). These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab × Ab^op × Ab^op → Ab.

What is the opposite group of an abelian group? In line with the first example, it should be the group itself with "reversed order" operation, i.e., the abelian group itself without modification to its group operation. But then the "op" markings are unecessary and misleading. --217.253.231.134 (talk) 09:14, 23 January 2012 (UTC)